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For stating this lemma, we need a definition. AuBCDEGH?IJKLMNPOX]^Z\_`abc d e f g ihjklmn(oqrwxy v)z*{+|,}-~./01?@ABCDEFGHIJKLMNOPQRSTUj3Circular chromatic indexes of graphs of large girth$G has large girth G is simpleklmnopqrstuv< ` ffff!` 33` 3fff` 3PPf` fff̙3f` 3f33` ff3fff` fff̙` ̙>?" dZ@,|?" dd@   " @ ` n?" dd@   @@``PT    @ ` `4p>> z8r811X8(  X3T  X " X  hBC"DEFJ.. e%/5BEMebebur7p_Uz?dM<H E"5 %% "$@               `" b b@ XC# n"l (I  X xBSCuDEFSuS @`"b X xBCbDEFbv1b @`"~@ X xBC1DEF1 @`"5p X 2BCDETF~UUaSHfup`R@@&^F pG` Mr_&x669D>2}uynpbaN@N@BJ 6%:| <@@                            `"Z  L b ) $  X# "    XB BCDEF00.%mR 9#-< HQXh  &-($!}tkc[!R$J)B.4=*P%g#~#&).bd@`"}  l  XB .BCDEF::mpv!4Li}unf\RH=1% !+5?IS[dlt{o_O. ~wmvx@`") /O d  XB &BCDEdFn^i s{2Not^<1& .?Op[Sa.^48@`"y[ t  XB 6BuCDElFvK$./B>TPfgtuxsklZcJY:N-C$: EOYbimqrK8<@`"9&   XB BCMDE4F>  ,< M5% @`"!? 6v Nb  XC# *J"O $ X NcFBC4DExF ,Hc}+E=lSj 4gWpKQD5?8,)+D2 ,>@@`" X cFB/CDE<FF7e-//-g)G$4"  $@`"= A, X cFBWC(DEHFRWMD :/%  !*4@!J(W&(@`"b b@ X# $"G X xBSCuDEFSuS @`"b X xBCbDEFbv1b @`"~@ X xBC1DEF1 @`"5b b@ X# bS"z X xBSCuDEFSuS @`"b X xBCbDEFbv1b @`"~@ X xBC1DEF1 @`"5\ b@ X "/o X xBSCuDEFSuS @`"b X xBCbDEFbv1b @`"~@ X xBC1DEF1 @`"5V X  BCDEFY+ "'*33:tXRoJ=%   (Fen(<Pik{ Bm#)4|9|@wDpI_L_LEZwH3 ~wrmold^T={(cW H=1%  X\@                                          `"W^Rd  X & BmCDEdFn% :N,`Gklmic]WQJB:5~5o8`;P8>0+48@`"L v $ !X  B6C(DEDFN  &.652!,&%(' $(@`" L3$ "X BvCDEDFN!&36HC^OwW^[vuhwRE/$(@`" | #X >BCDEpFzg}~oy\oDg)^Z]H FB&B3CAFNNQJQ:R%RPMLKLO Yg:<@`" $X B/CVDE,F6 %%;VRI!?*3/$.%@`"m/ %X BCDEF-- )eC2~ZGJl;0{MRf%}Z(e@kRAl8D7 =F \`@`"  &X fgBCDEF!!2K bw&3CVjpaTH=}4k+X!E/ DH@`"l 'X NgB|CnDExFM|lxkkiYeDc-ab fnbYRL$H/G;GHJIGF8C&A?=;99;BM>@@`"gd (X &gBmCDEdFn% :N,`Gklmic]WQJB:5~5o8`;P8>0+48@`"> )X gB.C^DE4F> &> O^ XP E(9-,.* @`"*2$ *X gB6C(DEDFN  &.652!,&%(' $(@`"I +X BhCDEF&&  '):5S?lIMPLC6;WroE# )tR+'L}m4VO1`hdUX7'NP@`"vh ,X fgBTCDEFAAtr hU>! -tETe7 :| !D q*AOT[SI6 >ghiG c1p]UK"A=?GTyee{TE90(!1St@`"8T\ -X S |`j "A`} j L cN NN}/kGrjL#j_   .X C \cj " ` j |0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\   /X C  ij "] `} j b*0 0X C ,nj "`  j d* 0 1X C rj "] `} j d* 0B X s *޽h ? fff̙80___PPT10.(02 Balloons1 P 1100\0(  \<,T  \ " b  3 \# i"j \ fBTCDEFAAtr hU>! -tETe7 :| !D q*AOT[SI6 >ghiG c1p]UK"A=?GTyee{TE90(!1St@`"  \ fBCDEF!!2K bw&3CVjpaTH=}4k+X!E/ DH@`"   \ NB|CnDExFM|lxkkiYeDc-ab fnbYRL$H/G;GHJIGF8C&A?=;99;BM>@@`"7wLp d \ &BmCDEdFn% :N,`Gklmic]WQJB:5~5o8`;P8>0+48@`", \ B.C^DE4F> &> O^ XP E(9-,.* @`"lp $  \ B6C(DEDFN  &.652!,&%(' $(@`"7-  \ BCDEF**)%:5IFXZdqpx|~xm.\VC' 6] )'$!n#?):V9mUQ,,VX@`" ( 3d  \B &BCDEdFn^i s{2Not^<1& .?Op[Sa.^48@`"Y.   \ BCDEF//{:[It&m~An,$:OgR)LP5j+ W3\^]{`d@`"   \ NB!CDExF%EDn:4Pmm}=}j!OyR^#du<[&>@@`"2 ! t \B 6BuCDElFvK$./B>TPfgtuxsklZcJY:N-C$: EOYbimqrK8<@`"'   \B BCMDE4F>  ,< M5% @`" v \ \ BTCDE`Fj_]^ 3zEAemz&uN4l!T24@`" b b@ \# 5"%  M \ xBSCuDEFSuS @`"b \ xBCbDEFbv1b @`"~@ \ xBC1DEF1 @`"5b b@ \# z" f \ xBSCuDEFSuS @`"b \ xBCbDEFbv1b @`"~@ \ xBC1DEF1 @`"5b b@ \# &"  \ xBSCuDEFSuS @`"b \ xBCbDEFbv1b @`"~@ \ xBC1DEF1 @`"5b b@ \C# &".a& \ xBSCuDEFSuS @`"b \ xBCbDEFbv1b @`"~@  \ xBC1DEF1 @`"5b b@ !\C# Y"N   "\ xBSCuDEFSuS @`"b #\ xBCbDEFbv1b @`"~@ $\ xBC1DEF1 @`"5J %\ B^CvDEXF`k%S"RFCov^SXCm-NSi-0@"x &\ fBCDEF!!2K bw&3CVjpaTH=}4k+X!E/ DH@`"nf  '\ NB|CnDExFM|lxkkiYeDc-ab fnbYRL$H/G;GHJIGF8C&A?=;99;BM>@@`"Z p (\ B.C^DE4F> &> O^ XP E(9-,.* @`"(+  )\ B|CyDE@FJ|q cQ)?6)BJK `q)y<ySoeXt5|"$@`"Cy *\ BCDEF**)%:5IFXZdqpx|~xm.\VC' 6] )'$!n#?):V9mUQ,,VX@`"}  +\ B$CDEF$ @`"8$ ,\ C  "` ` j b*0 -\ C , "`   d* 0 .\ C   "] `}  d* 0 /\ Bh 3 3fԔ8c?"x HH   L cN NN}/kGrjL#j_   0\ C j " @0  N cN NN}/kGroRjL#j_ B \ s *޽h ? fff̙80___PPT10.(02 QK0   q(  x  c $0\c *P      <$! / M Circular chromatic index,H  0޽h ? 33___PPT10i.#V.X+D=' = @B +j[PK0 ld%-P(  P` P c $A5  ??  ` P c $A3 ??\  T   P <7  QG=(V,E): a graph$  P <(cW-  X an integer  Vz  # P S3,$@ 0`2 P 0S `2 P 0 0`2  P 0`2  P 0#`2  P 0#TB  PB c $D TB  P c $DTB P c $D TB PB c $DTB P c $D` P c $A9 ??  P <PpGw  Ek-coloring of G is  P <t :6  < such that  ` P c $A2 ??a !! 1   P <Py 3A P c $A4 ??\ @ `T 8 $D 0 P c $A1  ?? ( @ 8  $@ 0  P <} ,$   0 Z a real number  !P <=-,$  0 i A (circular)& f ` "P c $A6  ??:    #P c $A7  ??g78  $D  0 $P <(JP>j,$ 0 30 %P <&,$ 0 31 &P <\",$ 0 32 'P <,$ 0 50.5 (P <ĕM/,$ 0 51.5 )P <d {C,,$ 0 @A 2.5-coloring *P c $A8 ??68 $@ 0 +P <`aq ,$ 0 _ r-coloring of G is ,P c $A: ??~>8 $@ 0 -P c $A> ??\ -T 8 $D 0H P 0޽h ? fff̙DD___PPT10fD.`d+ <D*A' = @B D@' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =%(D ' =%(D' =4@BBBB%(D' =+4 8?`CB ppt_xBCB0-ppt_w/2B*Y3>B ppt_x<*PD ' =+4 8?XCB ppt_yBCB ppt_yB*Y3>B ppt_y<*PD' =1:Bhidden*o3>+B#style.visibility<*P%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D4' =%(D' =4@BB#BB%()?)?D0' =.7 BBBBB5M 0.0 0.0 L -0.25 0.0 E*3>*B ppt_xB ppt_y=0BB<*PDf' =%(D' =%(D' =A@BBBB0B@@%(D' =1:B true*3>EB=style.textDecorationUnderline= `B<*PD' =%(D5' =%(D' =A@BBBB0B@%(D' =+4 8?`CB ppt_xBCB0-ppt_w/2B*Y3>B ppt_x<*PD ' =+4 8?XCB ppt_yBCB ppt_yB*Y3>B ppt_y<*PD' =1:Bhidden*o3>+B#style.visibility<*P%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* PD' =%(D' =%(D6' =A@BB?BB0B%()?)?D}' =.E7 BBBBBWM -1.94444E-6 3.7037E-7 L 0.1316 -0.00185 *3>*B ppt_xB ppt_y=@0BBAApBBl=Bֹr<*PD' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!P%(D' =%(D ' =%(D' =4@BBBB%(D' =+4 8?`CB ppt_xBCB0-ppt_w/2B*Y3>B ppt_x<*"PD ' =+4 8?XCB ppt_yBCB ppt_yB*Y3>B ppt_y<*"PD' =1:Bhidden*o3>+B#style.visibility<*"P%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#PD" ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(P%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)P%(D ' =%(D ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**P%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*#P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,P%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*P%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-P%(++0+P ++0+P ++0+P ++0+P ++0+P ++0+ P ++0+!P ++0+$P ++0+%P ++0+&P ++0+'P ++0+(P ++0+)P ++0++P +FPK0 D<0  (      <ԇ< \ ,$ 0 m%The circular chromatic number of G is,&3    c $An ??O _8 $@ 0  <lۇO \o,$ 0 [){ r : G has a circular r-coloring }**  <߇O P.o,$ 0 5min  <ׇA9 kThe chromatic number of G is 2f `   c $A N??  N  < X&= min { k : G has a k-coloring }''`   c $A ??4     c $A ??} j 58 $D 0H  0޽h ? 33  ___PPT10j .#VY+D ' = @B DQ ' = @BA?%,( < +O%,( < +D' =%(D)' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D*' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bdiamond(in)*<3<* D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+  ++0+  ++0+  +PK0   Phm (  h h <W  P Why  circular ?wz j  h `  ,$D 0 h <@   30=r h <l  13N ?F  h jz `2 h 0   h <4` ? 11  h <l  S s  12  h <M ,$  14TB  h c $D] ] @TB  hB c $D a TB  h c $D 3 TB h c $D F TB h c $D% S <z   h  ,$@ 0TB h c $D7M  h < 30 h <(  3r h 0P] ,$D 02 h s * 7,$D 02 h 06 d ,$@ 0H h 0޽h ? 33___PPT10g.=V88+99D;' = @B D' = @BA?%,( < +O%,( < +DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*hD' =0l9 BBBB*<3<*h)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*hD' =0l9 BBBB*<3<*h)?D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-u6Bdiamond(in)*<3<*h+7;PK0 B:"8(    <x"W  P Why  circular ?'8 j  +`    <X'@   30=r  <0+  13@ ?F  )jz `2  0    </` ? 11  <3 S s  12  <2M ,$  14TB  c $D] ] @TB  B c $D a TB " c $D 3 TB # c $D F TB $ c $D% S l f  &:,$D   0  <P;r V Ox~y`  0: I % <H@jf  "|f(x)-f(y)|_r e" 1"" B ' < E  ,$  0 ZThe distance between p, p in the circle is ,.3" ( c $A G??s { 8 G$D  0 * <tK 4 ,$   0 Mf is a circular r-coloring if8   0 TB , c $D7M  - <O 30 . <S  3rX / 0P] R2 1 s * 7X2 2 06 d 2 3 Hf?"6@ NNN?Nc,$@ 02 4 Hf?"6@ NNN?N -,$@ 0+ 5 T4X ?"6@ NNN?N,$ 0 5p. 6 TT\ ?"6@ NNN?N 3?,$ 0 8p  8  `0e0e    BC.DE@F  A@ f o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.<yyL1Q V1K&ef @     s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab,$D  0H  0޽h ? 33}___PPT10].=V88+xDQ' = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D0 ' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*'D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-s6Bwipe(down)*<3<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<**D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<**Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*&++0+' ++0+* ++0+5 ++0+6 +P  . (     TM  ?"6@ NNN?N~_ V$Coloring = distribution of resources%%P  T  ?"6@ NNN?NQ9  \Circular coloring = distribution of resources of periodic nature]]H  0޽h ? fff̙80___PPT10.j0S! PK0   o(    <q| T"A distributed computation problem ##  <Hv] } HV: a set of computers  <hzP< p ID: a set of data files  c $A ?? \ 8 $D 0bz 'c       ,$D 0h   c $A ??'   h   c $A ??z c   H  0޽h ? fff̙___PPT10.ͨ+nmD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+PK0   { (  ^2  6< ^2  6ma^2  6,q ^2  6) ^2  6 ] pRB  @ s *D3 RB   s *D VRB  @ s *DV CRB   s *D3, pRB   s *DC,  Y  B  3a  B  3b  Bt9  < 3c  BАI=) i 3d  B," 3e   <̗c i ,$ 0 i7If x ~ y, then x and y cannot operate at the same time88  <(V d \,$ 0 q?If x ~ y, then x and y must alternate their turns in operation.@@H  0޽h ? fff̙\T___PPT104.Ҩ33+JD' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+ ++0+ + PK0  (    <ԧ k9Schedule the operating time of the computers efficiently.::-  <<*],$ 0 [Efficiency: (number of computers operating on the average)/(total number). \\4  <а * ,$ 0 bThe computer time is discrete: time 0, 1, 2, & 22H  0޽h ? fff̙___PPT10.Ө@'+H%yD[' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+p+0+ ++0+ + PK0 ]U@  (    <F  F1: coloring solution  <tFhf QColor the vertices of G with k= `  c $A ??mjZe   <ř : colors.    c $A ?? 8 $D 0   c $A ??\ T 8 $D 0H  0޽h ? fff̙___PPT10.ӨW+D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+UPK0 ~0!V(  2 5 <ϙ6 ]  5a2 6 <4ә]Jqm 5e2 7 <֙s   5b2 8 <ٙ 5d2 9 <Hݙ  5cLB ;@ c $DCM sLB < c $DmLB = c $D0 # LB >@ c $Dm P LB ? c $DY Y @ <X< * \ ,$ 0 OColor the graph with 3 colors A <W E ,$  0 30 B <z ,$  0 31 C < ##  C,$  0 30 D <|,$  0 31 E <$MdRm,$ 0 32 F <M <  G <D  ,$ 0 9At time H <  ,$  0 30 I <0PV,$  0 BOperate machines J <h 9?,$  0 5a c K <   ,$  0 31 L <I=) O,$  0 5b d M < ,$ 0 32 N <\" { B,$ 0 3e O <4 ,$ 0 33 Q < 2 8,$ 0 5a c R <L"  ,$ 0 34 S <(I&O,$ 0 5b d T <, ~l ,$ 0 35 U <41 ,$ 0 3e V < 5,$ 0 HThe efficiency is 1/3H  0޽h ? fff̙<<___PPT10v<.;W+n+D7' = @B Dm7' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =%(Do' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*AD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*AD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*BD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*BD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*CD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*CD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ED' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ED2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =-u6Bdiamond(in)*<3<*V++0+@ ++0+A ++0+B ++0+C ++0+D ++0+E ++0+G ++0+H ++0+I ++0+J ++0+K ++0+L ++0+M ++0+N ++0+O ++0+Q ++0+R ++0+S ++0+T ++0+U ++0+V +PK0 `<n(  < < <Dn c1In general, the coloring solution has efficiency22_8 hh  <V00  < <sh`   3 h < c $A ??0 h  H < 0޽h ? 33___PPT10i.]R ;+D=' = @B +CPK0   ` ! (    <x3S Q2: Computer scientists solution 8    `2  0]  `2  0D`2  0 Ts `2  0` w `2   0 \ TB  B c $D TB   c $D&\ TB   c $D&\ TB  c $D  TB  c $D    0 F },$D 0  0  m ,$@ 0  0,,$D 0  0   ,$@ 0  0F } ,$@ 02  s *f] #,$D 02  s *fF  ,$@ 0X2  0ZX2  0w`   0 J ,$@ 0  0Iq,$@ 02  s *fZ,$@ 0  0 ,$@ 02 ! s *fw` ,$@ 0H  0޽h ? fff̙55___PPT105.Jg+KD5' = @B D_5' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dt' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?Dh ' =%(D ' =%(D' =4@BB1BB%()?)?Ds' =.;7 BBBBBMM -1.66667E-6 0.0 L -0.09844 0.07361 *3>*B ppt_xB ppt_y=@0BBAApBBIBl=<*D-' =4@BB1BB%()?)?D' =.I7 BBBBB[M -3.33333E-6 3.33333E-6 L 0.02361 -0.14699 *3>*B ppt_xB ppt_y=@0BBAApBBlA<Bl<*D/' =4@BB8BB%()?)?D' =.K7 BBBBB]M 8.33333E-7 -2.59259E-6 L -0.13767 -0.00787 *3>*B ppt_xB ppt_y=@0BBAApBB'B<*D%' =4@BB*BB%()?)?Dy' =.A7 BBBBBSM 3.05556E-6 -0.01064 L 0.08281 0.05225 *3>*B ppt_xB ppt_y=@0BBAApBB>)=B=<*D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(Dp ' =%(D ' =%(D+' =4@BB*BB%()?)?D' =.G7 BBBBBYM -3.05556E-6 -1.96532E-6 L 0.02761 0.1311 *3>*B ppt_xB ppt_y=@0BBAApBB `<B=<*D!' =4@BB@BB%()?)?Du' =.=7 BBBBBOM 0.00381 -0.00508 L 0.10225 -0.08901 *3>*B ppt_xB ppt_y=@0BBAApBB>I=B ],<*D-' =4@BB@BB%()?)?D' =.I7 BBBBB[M 5.55556E-7 -3.87283E-6 L -0.08281 -0.0682 *3>*B ppt_xB ppt_y=@0BBAApBB)B) <*D/' =4@BB*BB%()?)?D' =.K7 BBBBB]M -4.44444E-6 -4.04624E-6 L -0.02361 0.14174 *3>*B ppt_xB ppt_y=@0BBAApBBlAB"=<*+.PK0 |` Ql (  l l 0VJf,$D 0>L z l# F9Z2  l s *zZ2  l s *?Z2  l s *VZ2  l s **`Z2  l s *`^B l 6D>^B l 6D> ^B l@ 6D>C^B l 6D>YY^B l 6D> Y2 l 6f>,$@ 02 l 6f>$,$D 0>L z l# 6 Z2 l s *zZ2 l s *?Z2 l s *VZ2 l s **`Z2 l s *`B l@ 6D>c ,$@ 0B l 6D>c ,$D 0B l 6D>9 `,$@  0B l@ 6D>Iz 9 I,$D  0^B  l 6D>M I2 !l 6f>s&7 l,$@  02 "l 6f>F9 J ?,$D 0XB Ml 0D>z XB Nl 0D> XB Ol@ 0D>9 JXB Pl 0D>`d P `H l 0޽h ? fff̙___PPT10.FV{M+ D' = @B DE' = @BA?%,( < +O%,( < +D@' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*l%(Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*l%(Dp' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*MlD' =1:Bhidden*o3>+B#style.visibility<*Ml%(D)' =4@BB BB%(D' =-g6B fade*<3<*NlD' =1:Bhidden*o3>+B#style.visibility<*Nl%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-g6B fade*<3<*lD+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-g6B fade*<3<*lD' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*OlD' =1:Bhidden*o3>+B#style.visibility<*Ol%(DY' =4@BB BB%()))D' =-g6B fade*<3<*PlD' =1:Bhidden*o3>+B#style.visibility<*Pl%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-g6B fade*<3<*lD+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-g6B fade*<3<*lD@' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*!l%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*"l%(+(<PK0 "  `Lg|(  |X *| 0VJf +| 0@ j,$@ 02 ,| <'m ,$@ 0" -| 0 `,$@  0B .| <v / ,$D 08 F9 6|F92@ z |F9Z2 | s *zZ2 | s *?Z2  | s *VZ2  | s **`Z2  | s *``B /| 0D>`B 0| 0D> `B 2|B 0D>C`B 3| 0D>YY`B 4| 0D> Y^2 5| 6f>^2 7| 6f>$8 6  @|6 @N z | 6 Z2 | s *zZ2 | s *?Z2 | s *VZ2 | s **`Z2 | s *``B ;|B 0D>c `B <| 0D> c `B =| 0D>9  ``B >|B 0D>z I9 I`B ?| 0D>M I^2 A| 6f>s&7 l^2 B| 6f>F9 J ?l Y I|Y,$@ 0@N z | YZ2 | s *zZ2 | s *?Z2 | s *VZ2 | s **`Z2 | s *``B C| 0D>F`B E| 0D>]`B F| 0D>F`B G|B 0D>``B H|B 0D>`2 V| 6f>@Q,$@ 02 W| 6f>,$D 0l   Y ]| Y,$@ 0@N z |   YZ2 | s *zZ2  | s *?Z2 !| s *VZ2 "| s **`Z2 #| s *``B X|B 0D> /  `B Y| 0D>/ m `B Z| 0D> w `B [| 0D>w `B \|B 0D>c 2 ^| 6f>? P a 8 ,$@ 02 _| 6f>,$D  0l  S+ e| S+,$@  0@N z $|  S+Z2 %| s *zZ2 &| s *?Z2 '| s *VZ2 (| s **`Z2 )| s *``B `| 0D>/  `B a| 0D>m `B b|B 0D>`B c| 0D> &`B d| 0D> ? 2 f| 6f>  ,$@  02 g| 6f>v*;o,$D  0H | 0޽h ? fff̙___PPT10n.FV{M+0aDB' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*+|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*I|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*V|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*W|%(D' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*,|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*]|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*^|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*_|%(D' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*-|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*e|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*f|%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*g|%(Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*.|%(+PK0 (  8 W fn  <0W-M g%The efficiency of this scheduling is &&$h  c $A "??6 "  <áF 9f  S!Better than the coloring solution""H  0޽h ? fff̙___PPT10i.HV +D=' = @B +PK0  3(    <LɡG1  S3: Circular coloring solution  <͡a <Let r=`  c $A$ #??#[ #  <ҡa U#Let f be a circular r-coloring of G$$  <֡&2 +x operates at time k iff for some integer m,,`   c $A% $??~  $H  0޽h ? fff̙___PPT10i.JV0~u+D=' = @B +??PK0 a(  Ll a   a ,$D 02  <ݡ]=  >   <ܡ M m 30  <@ , 31  <  32  <q% _E  33   <=+  34   <`a 7r=4.48   BCDE4FA> Dq)PDwu @    S"  ,$D 0  <t  ,$  0 70H   BCDE@F A>-[ :lz@m[6p @     S" w  ,$D 0  <Z z ,$  0 71h   BClDEXF(A>k!42^elxY)NC`G!AkL&D@       S"  x3 0 ,$D 0  <Dc  ,$D  0 728   BCDE4FA> P2[ @"dDOP' @    S" ! ,$D  0  <$@`,$   0 738   BCDE4FA> qJH=vc @    S" ,$D  0  <8 ,$D   0 74X   BICQDELF$A> .DLm='(6l>FIa>3=Q@      S"  ,$D  0  0,$  0 55 25  T ?"6@ NNN?NV Wv,$ 0 ? Efficiency =   c $A !?? j* 8 !$D 0H  0޽h ? fff̙))___PPT10).KVP٘+?aD(' = @B DM(' = @BA?%,( < +O%,( < +D' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B+checkerboard(across)*<3<* D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bdiamond(in)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D*' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%( D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D' =%( DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+ ++0+ +PK0   u (  F ~J8  ~J8  <*~) W%The efficiency of this scheduling is &&h  c $A" ??J8 l M)  M) ,$D 0`2   0  g`2   0=D`2   0M  `2   0 J `2  09 c TB B c $D# TB  c $D # TB B c $Dg  TB  c $Dz TB  c $Dz g   <p2= ), 3b  <5Mg3 3e  T9 ?"6@ NNN?NQ 8  5a  TH< ?"6@ NNN?NZ < 2 \  5c  T? ?"6@ NNN?N  5dM  T0C ?"6@ NNN?Nm h,$ 0 W%For this example, the efficiency is &&  c $A ?? 8 $D 0H  0޽h ? fff̙___PPT10{.LVf1+>GD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +PK0  (  8 ~!   ~!   <N~" tTheorem [Yeh-Zhu, submitted]f   < T#C X&The optimal scheduling has efficiency ''h   c $A' 8?? |  8h   c $A" 9??  !  9H  0޽h ? fff̙___PPT10i.LVf1+D=' = @B + PK0 d\  (  S  0Y=H@8___PPT9 #Vince, 1988. star chromatic numberNff8X`  c $A + ??W  + 2  <kN,$ 0 DAbout 100 papers published, and publications are accelerating.2Ef<z  _    ,$D 0G   <[ 2 @U 0U 0 %Quote from Feder, Hell, Mohar (2003):&&,     <t=s _ r, The theory of circular colorings of graphs has become an important branch of chromatic graph theory with many exciting results and new techniques. *ffH  0޽h ? fff̙___PPT10.+uD' = @B DF' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+ +<PK0 y q  (    <n  FQuestions concerning`  c $A ??)    B܃  CRelation between f  s *A ??     Bp q Band other graph    B䌨f = parameters: f   s *A I??Px I   B8k 41.l u  u ,$D 0   <,  8Circular chromatic number of special classes of graphs: planar graphs, graphs with forbidden minor, line graphs, distance graphs, &    <(u  42.l  $   $ ,$D 0  <x  G3. How to calculate h  c $A ?? $  Z  T ?"6@ NNN?N,$ 0 d24. Generalization to circular coloring of digraphs33H  0޽h ? fff̙SK___PPT10+.8O +: bD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +PK0 . & pp (  p p <Tn  FQuestions concerning` p c $A J??)  JF u[   p u[   p <[  \*Circular chromatic number of line graphs.++  p <u  42.F  $   p  $  p <  G3. How to calculate h p c $A M?? $  M8 kq pkq p <  CRelation between h p c $A K??   K p < q Band other graph  p <Ũf = parameters: h p c $A L??Px L  p <0Ĩk 41.@ 2p pf/ p T0̨ ?"6@ NNN?N2 R e and circular flow numberh p c $A ?? `p H p 0޽h ? fff̙___PPT10.8O +D' = @B DS' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<* pD' =1:Bhidden*o3>+B#style.visibility<* p%(DY' =4@BB BB%()))D' =-g6B fade*<3<*pD' =1:Bhidden*o3>+B#style.visibility<*p%(+X P o g $ $(  $ $ TԨ ?"6@ NNN?N  OAn equivalent definition for ` $ c $A &??  &) $ T٨ ?"6@ NNN?N Z  g p, q: positive integers.1 $ T@ި ?"6@ NNN?Nr o#A (p, q)-coloring of G is a mapping0$m $ TDب ?"6@ NNN?N   ) f: V {0, 1, ... , p-1} such thatL* )B  $ TD?"0@NNN?N 3C   $ T ?"6@ NNN?N : ?  ] x ~ y   $ N?"6@ NNN?Ns = `  $ c $A &??/ G '  &`  $ c $A' &?? |  &H $ 0޽h ? fff̙___PPT10i.ie@O+D=' = @B + P   $ (m (  ( ( Tl ?"6@ NNN?N  OAn equivalent definition for ` ( c $A &??  &) ( T ?"6@ NNN?N Z  g p, q: positive integers.1 ( T ?"6@ NNN?Nr o#A (p, q)-coloring of G is a mapping0$m ( T8 ?"6@ NNN?N   ) f: V {0, 1, ... , p-1} such thatL* )B ( TD?"0@NNN?N 3C  ( T ?"6@ NNN?N : ?  ] x ~ y   ( N?"6@ NNN?Ns = `  ( c $A &??/ G '  &`  ( c $A' &?? |  &f  ( s *A &?? k &H ( 0޽h ? fff̙___PPT10i.ie@O+D=' = @B +P $\(    T ?"6@ NNN?NA Z(Circular chromatic index of cubic graphs))2  N?"6@ NNN?NV  2  N?"6@ NNN?N,-2  N?"6@ NNN?N jf 2  N?"6@ NNN?N2   N?"6@ NNN?N =9 2   N?"6@ NNN?N A 2   N?"6@ NNN?N  S 2   N?"6@ NNN?Np a 2   N?"6@ NNN?N/ G 2  N?"6@ NNN?N 4 B @ ND?"0@NNN?N YB  ND?"0@NNN?N B  ND?"0@NNN?N" " B @ ND?"0@NNN?Nj B  ND?"0@NNN?N &B @ ND?"0@NNN?N0 F B  ND?"0@NNN?NG \ B @ ND?"0@NNN?N  B  ND?"0@NNN?N%   B  ND?"0@NNN?Nw)  B  ND?"0@NNN?N B  ND?"0@NNN?N B  ND?"0@NNN?NpJ &B  ND?"0@NNN?N\ 3  B @ ND?"0@NNN?N    T ?"6@ NNN?N,  < H  0޽h ? fff̙___PPT10i.ge R+D=' = @B +%P $$$,,,E$(  , , Tx" ?"6@ NNN?NA Z(Circular chromatic index of cubic graphs))2 , N?"6@ NNN?NV  2 , N?"6@ NNN?N,-2 , N?"6@ NNN?N jf 2 , N?"6@ NNN?N2 , N?"6@ NNN?N =9 2 , N?"6@ NNN?N A 2  , N?"6@ NNN?N  S 2  , N?"6@ NNN?Np a 2  , N?"6@ NNN?N/ G 2  , N?"6@ NNN?N 4 B  ,@ ND?"0@NNN?N YB , ND?"0@NNN?N B , ND?"0@NNN?N" " B ,@ ND?"0@NNN?Nj B , ND?"0@NNN?N &B ,@ ND?"0@NNN?N0 F B , ND?"0@NNN?NG \ B ,@ ND?"0@NNN?N  B , ND?"0@NNN?N%   B , ND?"0@NNN?Nw)  B , ND?"0@NNN?N B , ND?"0@NNN?N B , ND?"0@NNN?NpJ &B , ND?"0@NNN?N\ 3  B ,@ ND?"0@NNN?N   , T1 ?"6@ NNN?N,  <  , T4 ?"6@ NNN?N*   38 , T8 ?"6@ NNN?N  30 , T/ ?"6@ NNN?N v  33  , T`: ?"6@ NNN?N %  30 !, TA ?"6@ NNN?N 0  36 ", TE ?"6@ NNN?NG5 %  32 #, Td@ ?"6@ NNN?N {i  35 $, TK ?"6@ NNN?N    31 %, TP ?"6@ NNN?NF f  39 &, TN ?"6@ NNN?N j X?  32 ', TV ?"6@ NNN?N   37 (, TP[ ?"6@ NNN?N m (  410 ), TlY ?"6@ NNN?NC c  35 *, Ta ?"6@ NNN?N6 V  38 +, Te ?"6@ NNN?N  34  ,, Td ?"6@ NNN?Nfq/ KA (11,3)-coloring of L(P)H , 0޽h ? fff̙___PPT10i.ge R+D=' = @B +oP ~$ 0(  0E 0 Tm ?"6@ NNN?N*I QQuestion: Is it true that every 2-edge connected cubic graph hasRR` 0 c $A &??F= N &I 0 Tr ?"6@ NNN?NC *"I  UProblem: Prove that every 2-edge connected cubic planar graph G has VV` 0 c $A &??   & 0 Tf j Hamed Hatami $8  4 ZГ ?"6@ NNN?N9eY pRuzbeh Tusserkani,$   4 Z8 ?"6@ NNN?N9pY W Xuding Zhu] H 4 0޽h ? fff̙___PPT10i.ke`+D=' = @B +!P   % &8t (  80 8 TĞ ?"6@ NNN?Nw  nPeyman Afshani, $ 8 T, ?"6@ NNN?Nq* ; Theorem [ 1 8 T ?"6@ NNN?N&  oMahsa Ghandehari,$ 1 8 T0 ?"6@ NNN?NdT  oMahya Ghandehari,$ 2 8 Z, ?"6@ NNN?NF,>f j Hamed Hatami $8 8 Z ?"6@ NNN?N9eY pRuzbeh Tusserkani,$  8 Zغ ?"6@ NNN?N9pY W Xuding Zhu] 2  8 N?"6@ NNN?N -O 2  8 N?"6@ NNN?NI g 2  8 N?"6@ NNN?NB -2  8 N?"6@ NNN?NS`2  8 N?"6@ NNN?N `B 8 ND?"0@NNN?NO ,jB 8@ ND?"0@NNN?N9 P IB 8 ND?"0@NNN?NIP vB 8 ND?"0@NNN?NIIB 8@ ND?"0@NNN?NO B 8 ND?"0@NNN?NP YB 8@ ND?"0@NNN?NI,B2 8 N?"6@ NNN?N 022 8 N?"6@ NNN?N  2 8 N?"6@ NNN?NmB 8 ND?"0@NNN?N 0B 8 ND?"0@NNN?N0B  8@ ND?"0@NNN?N  !8 S T0e0e    BCDEF A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||\T& 'O@  s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab[ 0 ` "8 c $A &??Y a &` #8 c $A &??,@X4 &f $8 s *A &?? M} _  & %8 H?"6@ NNN?N 9  f &8 s *A &??% =-  &H 8 0޽h ? fff̙___PPT10i.ke`+D=' = @B +} P )$$(  $r $ S Ƕ-XA`}   r $ S Ƕ.X `  H $ 0޽h ? fff̙___PPT10i.hQ/+D=' = @B +P P%#D;(  Df D c 0A lineP"`Ct D N<Ѷ?"6@ NNN?Nm 31 D N(Ӷ?"6@ NNN?N5 32 D N׶?"6@ NNN?Nm` N 30 D Nڶ?"6@ NNN?NL l  30 D NH߶?"6@ NNN?N  32 D N?"6@ NNN?N0~ l P 33 D N?"6@ NNN?NM; 30 D N?"6@ NNN?N}~l 31 D N8?"6@ NNN?NI i  31 D N?"6@ NNN?N< =+\  30 D N8?"6@ NNN?N]  }  32 D N?"6@ NNN?NF f  31 D N?"6@ NNN?N  30  D NT?"6@ NNN?N $ 33 "D Td ?"6@ NNN?Nf ZA 4-coloring f0  #D NT?"6@ NNN?NIdRi  32H D 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +"P og)((  (f ( c 0A lineP"`Ct ( N?"6@ NNN?Nm 31 ( N ?"6@ NNN?N5 32 ( N?"6@ NNN?Nm` N 30 ( Nx?"6@ NNN?NL l  30 ( N8?"6@ NNN?N  32 ( N!?"6@ NNN?N0~ l P 33  ( N%?"6@ NNN?NM; 30  ( N)?"6@ NNN?N}~l 31  ( N8(?"6@ NNN?NI i  31  ( N/?"6@ NNN?N< =+\  30  ( N(4?"6@ NNN?N]  }  32 ( N2?"6@ NNN?NF f  31 ( N8;?"6@ NNN?N  30 ( N:?"6@ NNN?N $ 33 ( TA ?"6@ NNN?Nf ZA 4-coloring f0  ( N@G?"6@ NNN?NIdRi  32" ( TdK ?"6@ NNN?Nw{c `.We define a digraph induced by this coloring// ( s *A (?33?]@8 ($D 0H ( 0޽h ? fff̙[S___PPT103.ne0 W2+iGRD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =-g6B fade*<3<*(+P `%H(  Hf H c 0A lineP"`Ct H NQ?"6@ NNN?Nm 31 H NU?"6@ NNN?N5 32 H NhY?"6@ NNN?Nm` N 30 H N]?"6@ NNN?NL l  30 H N`?"6@ NNN?N  32 H Nd?"6@ NNN?N0~ l P 33  H NXh?"6@ NNN?NM; 30  H Nl?"6@ NNN?N}~l 31  H No?"6@ NNN?NI i  31  H Nps?"6@ NNN?N< =+\  30  H NHw?"6@ NNN?N]  }  32 H N{?"6@ NNN?NF f  31 H NC?"6@ NNN?N  30 H NH?"6@ NNN?N $ 33 H T ?"6@ NNN?Nf ZA 4-coloring f0 f H s *A &??Z X &B H TD?"0@NNN?NG G   H T  ?"6@ NNN?Nj H0 1 2 3 0B H ZDԔ?"0@NNN?N Sz B H ZDԔ?"0@NNN?N  B H ZDԔ?"0@NNN?N m B H ZDԔ?"0@NNN?N C  H N@?"6@ NNN?NIdRi  32H H 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +P p%!Lk(  Lf L c 0A lineP"`Ct L N<?"6@ NNN?Nm 31 L N?"6@ NNN?N5 32 L N?"6@ NNN?Nm` N 30 L N?"6@ NNN?NL l  30 L N<?"6@ NNN?N  32 L N?"6@ NNN?N0~ l P 33  L NȪ?"6@ NNN?NM; 30  L N?"6@ NNN?N}~l 31  L N?"6@ NNN?NI i  31  L N`?"6@ NNN?N< =+\  30  L N0?"6@ NNN?N]  }  32 L NL?"6@ NNN?NF f  31 L N?"6@ NNN?N  30 L N?"6@ NNN?N $ 33 L T ?"6@ NNN?Nf ZA 4-coloring f0 f L s *A &??Z X &B L TD?"0@NNN?NG G B L ZDԔ?"0@NNN?N t`,$D  0B L@ ZDԔ?"0@NNN?N,$D  0 L N?"6@ NNN?NIdRi  32  L T ?"6@ NNN?Nj H0 1 2 3 0B L ZDԔ?"0@NNN?N Sz B L ZDԔ?"0@NNN?N  B  L ZDԔ?"0@NNN?N m B !L ZDԔ?"0@NNN?N C H L 0޽h ? fff̙PH___PPT10(.ne0 W2+7aD' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bwipe(up)*<3<*LD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bwipe(up)*<3<*L+$P UM%#P(  Pf P c 0A lineP"`Ct P N?"6@ NNN?Nm 31 P N?"6@ NNN?N5 32 P Nd?"6@ NNN?Nm` N 30 P N?"6@ NNN?NL l  30 P N?"6@ NNN?N  32 P Nx?"6@ NNN?N0~ l P 33  P N?"6@ NNN?NM; 30  P N0?"6@ NNN?N}~l 31  P N?"6@ NNN?NI i  31  P N?"6@ NNN?N< =+\  30  P N?"6@ NNN?N]  }  32 P ND?"6@ NNN?NF f  31 P N?"6@ NNN?N  30 P N4?"6@ NNN?N $ 33 P T  ?"6@ NNN?Nf ZA 4-coloring f0 f P s *A &??Z X &B P TD?"0@NNN?NG G B P ZDԔ?"0@NNN?N t`B P@ ZDԔ?"0@NNN?NB P ZDԔ?"0@NNN?N ,$D  0B P ZDԔ?"0@NNN?N ,$D  0B P@ ZDԔ?"0@NNN?N&,$D 0 P N?"6@ NNN?NIdRi  32  P T ?"6@ NNN?Nj H0 1 2 3 0B  P ZDԔ?"0@NNN?N Sz B !P ZDԔ?"0@NNN?N  B "P ZDԔ?"0@NNN?N m B #P ZDԔ?"0@NNN?N C H P 0޽h ? fff̙I A ___PPT10! .ne0 W2+k*D ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-u6Bwipe(right)*<3<*PD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-s6Bwipe(down)*<3<*PD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-s6Bwipe(down)*<3<*P+*P %"'X!(  Xf X c 0A lineP"`Cy X N"?"6@ NNN?Nm 31 X N@&?"6@ NNN?N5 32 X N)?"6@ NNN?Nm` N 30 X N,?"6@ NNN?NL l  30 X N1?"6@ NNN?N  32 X N4?"6@ NNN?N0~ l P 33  X Nt7?"6@ NNN?NM; 30  X N:?"6@ NNN?N}~l 31  X N,>?"6@ NNN?NI i  31  X NA?"6@ NNN?N< =+\  30  X ND?"6@ NNN?N]  }  32 X N@H?"6@ NNN?NF f  31 X NK?"6@ NNN?N  30 X NN?"6@ NNN?N $ 33 X TR ?"6@ NNN?Nf ZA 4-coloring f0 f X s *A &??Z X &B X TD?"0@NNN?NG G B X ZDԔ?"0@NNN?N t`B X@ ZDԔ?"0@NNN?NB X ZDԔ?"0@NNN?N B X ZDԔ?"0@NNN?N B X@ ZDԔ?"0@NNN?N&B X ZDԔ?"0@NNN?Nna\ ,$D  0B X ZDԔ?"0@NNN?Nn I,$D  0B  X ZDԔ?"0@NNN?NW 2,$D  0B !X ZDԔ?"0@NNN?N C,$D  0 "X Np[?"6@ NNN?NIdRi  32  #X T^ ?"6@ NNN?Nj H0 1 2 3 0B $X ZDԔ?"0@NNN?N Sz B %X ZDԔ?"0@NNN?N  B &X ZDԔ?"0@NNN?N m B 'X ZDԔ?"0@NNN?N C H X 0޽h ? fff̙  ___PPT10 .ne0 W2+~D ' = @B D} ' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-s6Bwipe(down)*<3<* XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!X%(D' =-s6Bwipe(down)*<3<*!X+&P %%%--d$(  df d c 0A lineP"`Ct d Nd?"6@ NNN?Nm 31 d N?"6@ NNN?N5 32 d N?"6@ NNN?Nm` N 30 d Nx?"6@ NNN?NL l  30 d NԺ?"6@ NNN?N  32 d N?"6@ NNN?N0~ l P 33  d N\?"6@ NNN?NM; 30  d N4?"6@ NNN?N}~l 31  d N ?"6@ NNN?NI i  31  d N|?"6@ NNN?N< =+\  30  d NL?"6@ NNN?N]  }  32 d N$?"6@ NNN?NF f  31 d N?"6@ NNN?N  30 d Nl?"6@ NNN?N $ 33 d T, ?"6@ NNN?Nf ZA 4-coloring f0 f d s *A &??Z X &B d TD?"0@NNN?NG G B d ZDԔ?"0@NNN?N t`B d@ ZDԔ?"0@NNN?NB d ZDԔ?"0@NNN?N B d ZDԔ?"0@NNN?N B d@ ZDԔ?"0@NNN?N&B d ZDԔ?"0@NNN?Nna\ B d ZDԔ?"0@NNN?Nn IB d ZDԔ?"0@NNN?NW 2B d ZDԔ?"0@NNN?N CB d@ ZDԔ?"0@NNN?N`B d ZDԔ?"0@NNN?N&B d ZDԔ?"0@NNN?N&   d N(?"6@ NNN?NIdRi  32  !d T8 ?"6@ NNN?Nj H0 1 2 3 0B "d ZDԔ?"0@NNN?N Sz B #d ZDԔ?"0@NNN?N  B $d ZDԔ?"0@NNN?N m B %d ZDԔ?"0@NNN?N C B &d ZDԔ?"0@NNN?N  B 'd@ ZDԔ?"0@NNN?N? z ? B (d ZDԔ?"0@NNN?N& B )d ZDԔ?"0@NNN?N@ B *d@ ZDԔ?"0@NNN?N  `B +d@ ZDԔ?"0@NNN?N" Z B ,d ZDԔ?"0@NNN?Nj B -d ZDԔ?"0@NNN?N  H d 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +x!P   %'-l (  l l N?"6@ NNN?Ntb 31 l N?"6@ NNN?N5 32 l N?"6@ NNN?NV v 30 l ND?"6@ NNN?NL l  30 l N?"6@ NNN?N y  32 l N4?"6@ NNN?N y  33  l N ?"6@ NNN?N)uI 30  l Nl?"6@ NNN?N% 31  l N?"6@ NNN?NI i  31  l N?"6@ NNN?N   30  l N?"6@ NNN?N   32 l NL!?"6@ NNN?N ~ l  31 l N(%?"6@ NNN?N ] K  30 l N#?"6@ NNN?N5 33 l T, ?"6@ NNN?Nf ZA 4-coloring f0 f l s *A &??Z X &B l TD?"0@NNN?NG G B l ZDԔ?"0@NNN?N t`B l@ ZDԔ?"0@NNN?NB l ZDԔ?"0@NNN?N B l ZDԔ?"0@NNN?N B l@ ZDԔ?"0@NNN?N&B l ZDԔ?"0@NNN?Nna\ B l ZDԔ?"0@NNN?Nn IB l ZDԔ?"0@NNN?NW 2B l ZDԔ?"0@NNN?N CB l@ ZDԔ?"0@NNN?N`B l ZDԔ?"0@NNN?N&B l ZDԔ?"0@NNN?N&   l N(1?"6@ NNN?N 5  32B &l ZDԔ?"0@NNN?N  B 'l@ ZDԔ?"0@NNN?N? z ? B (l ZDԔ?"0@NNN?N& B )l ZDԔ?"0@NNN?N@ B *l@ ZDԔ?"0@NNN?N  `B +l@ ZDԔ?"0@NNN?N" Z B ,l ZDԔ?"0@NNN?Nj B -l ZDԔ?"0@NNN?N  H l 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +P  %<s(  < < N9?"6@ NNN?N]  5Lemma: If G has a 4-coloring f such that (1): is acyclic; (2): each directed path of contains at most 2 vertices of color 3, then 0f < s *A &??=  &f < s *A &??   &f < s *A &??P- x &H < 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +c,P ""&((p!(  p p N$D?"6@ NNN?Ntb 31 p NG?"6@ NNN?N5 32 p NJ?"6@ NNN?NV v 30 p N8N?"6@ NNN?NL l  30 p NQ?"6@ NNN?N y  32 p NT?"6@ NNN?N y  33 p NLX?"6@ NNN?N)uI 30  p N[?"6@ NNN?N% 31  p N_?"6@ NNN?NI i  31  p N`b?"6@ NNN?N   30  p Ne?"6@ NNN?N   32  p Ni?"6@ NNN?N ~ l  31 p Ntl?"6@ NNN?N ] K  30 p No?"6@ NNN?N5 33 p Tr ?"6@ NNN?Nf ZA 4-coloring f0  p s *A &??,8 &$@  0B p TD?"0@NNN?NG G B p ZDԔ?"0@NNN?N t`B p@ ZDԔ?"0@NNN?NB p ZDԔ?"0@NNN?N B p ZDԔ?"0@NNN?N B p@ ZDԔ?"0@NNN?N&B p ZDԔ?"0@NNN?Nna\ B p ZDԔ?"0@NNN?Nn IB p ZDԔ?"0@NNN?NW 2B p ZDԔ?"0@NNN?N CB p@ ZDԔ?"0@NNN?N`B p ZDԔ?"0@NNN?N&B p ZDԔ?"0@NNN?N&  p N}?"6@ NNN?N 5  32B  p ZDԔ?"0@NNN?N  B !p@ ZDԔ?"0@NNN?N? z ? B "p ZDԔ?"0@NNN?N& B #p ZDԔ?"0@NNN?N@ B $p@ ZDԔ?"0@NNN?N  `B %p@ ZDԔ?"0@NNN?N" Z B &p ZDԔ?"0@NNN?Nj B 'p ZDԔ?"0@NNN?N  M (p T܃ ?"6@ NNN?Np,$  0 W is acyclic  H p 0޽h ? fff̙  ___PPT10 .ne0 W2+Do ' = @B D* ' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(p+8+0+(p + P 5-&%(t(  t t Nd?"6@ NNN?Ntb 31 t N?"6@ NNN?N5 32 t Np?"6@ NNN?NV v 30 t N?"6@ NNN?NL l  30 t N?"6@ NNN?N y  33 t N0?"6@ NNN?N)uI 30  t N?"6@ NNN?N% 31  t N$?"6@ NNN?NI i  31  t N?"6@ NNN?N   30  t N?"6@ NNN?N   32  t N?"6@ NNN?N ~ l  31 t N?"6@ NNN?N ] K  30 t N?"6@ NNN?N5 33 t T| ?"6@ NNN?Nf ZA 4-coloring f0 f t s *A &??, &B t TD?"0@NNN?NG G B t ZDԔ?"0@NNN?N t`B t@ ZDԔ?"0@NNN?NB t ZDԔ?"0@NNN?N B t ZDԔ?"0@NNN?N B t@ ZDԔ?"0@NNN?N&B t ZDԔ?"0@NNN?Nna\ B t ZDԔ?"0@NNN?N CB t@ ZDԔ?"0@NNN?N`B t ZDԔ?"0@NNN?N&B t ZDԔ?"0@NNN?N&  t N?"6@ NNN?N 5  32B  t ZDԔ?"0@NNN?N  B !t@ ZDԔ?"0@NNN?N? z ? B "t ZDԔ?"0@NNN?N& B #t ZDԔ?"0@NNN?N@ B $t@ ZDԔ?"0@NNN?N  `B %t@ ZDԔ?"0@NNN?N" Z B &t ZDԔ?"0@NNN?Nj B 't ZDԔ?"0@NNN?N   (t T ?"6@ NNN?Np W is acyclic  H t 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +P  &"%xJ(  x x N?"6@ NNN?Ntb 31 x N?"6@ NNN?N5 32 x N?"6@ NNN?NV v 30 x N`?"6@ NNN?NL l  30 x N?"6@ NNN?N y  33 x N<?"6@ NNN?N)uI 30  x N?"6@ NNN?NI i  31  x N?"6@ NNN?N   30  x N?"6@ NNN?N   32  x Nh?"6@ NNN?N ~ l  31  x N8?"6@ NNN?N ] K  30 x NT?"6@ NNN?N5 33 x T ?"6@ NNN?Nf ZA 4-coloring f0 f x s *A &??, &B x TD?"0@NNN?NG G B x ZDԔ?"0@NNN?N t`B x@ ZDԔ?"0@NNN?NB x ZDԔ?"0@NNN?N B x ZDԔ?"0@NNN?N B x@ ZDԔ?"0@NNN?N&B x ZDԔ?"0@NNN?N CB x ZDԔ?"0@NNN?N&B x ZDԔ?"0@NNN?N&  x N?"6@ NNN?N 5  32B x ZDԔ?"0@NNN?N  B x@ ZDԔ?"0@NNN?N? z ? B x ZDԔ?"0@NNN?N& B  x ZDԔ?"0@NNN?N@ B !x@ ZDԔ?"0@NNN?N  `B "x@ ZDԔ?"0@NNN?N" Z B #x ZDԔ?"0@NNN?Nj B $x ZDԔ?"0@NNN?N   %x T ?"6@ NNN?Np W is acyclic  H x 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P ?70&"|(  | | NL?"6@ NNN?Ntb 31 | N?"6@ NNN?N5 32 | N?"6@ NNN?NV v 30 | N?"6@ NNN?NL l  30 | N#?"6@ NNN?N y  33 | N!?"6@ NNN?N)uI 30 | N+?"6@ NNN?NI i  31  | N.?"6@ NNN?N   30  | N`0?"6@ NNN?N   32  | Nl*?"6@ NNN?N ~ l  31  | Np8?"6@ NNN?N5 33 | T0; ?"6@ NNN?Nf ZA 4-coloring f0 f | s *A &??, &B | TD?"0@NNN?NG G B | ZDԔ?"0@NNN?N t`B |@ ZDԔ?"0@NNN?NB | ZDԔ?"0@NNN?N B |@ ZDԔ?"0@NNN?N&B | ZDԔ?"0@NNN?N CB | ZDԔ?"0@NNN?N&B | ZDԔ?"0@NNN?N&  | N`C?"6@ NNN?N 5  32B | ZDԔ?"0@NNN?N  B |@ ZDԔ?"0@NNN?N? z ? B | ZDԔ?"0@NNN?N& B | ZDԔ?"0@NNN?N@ B |@ ZDԔ?"0@NNN?N" Z B  | ZDԔ?"0@NNN?Nj B !| ZDԔ?"0@NNN?N   "| T(I ?"6@ NNN?Np W is acyclic  H | 0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +P @&T(    NJ?"6@ NNN?Ntb 31  NHT?"6@ NNN?N5 32  NR?"6@ NNN?NV v 30  NZ?"6@ NNN?NL l  30  N(^?"6@ NNN?N y  33  Na?"6@ NNN?N)uI 30  Nd?"6@ NNN?NI i  31   N+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*+p+0+  ++0+  +$P ##@'*/W#(    NXR?"6@ NNN?Ntb 31  N4V?"6@ NNN?N5 32  NY?"6@ NNN?NV v 30  N\?"6@ NNN?NL l  30  N`?"6@ NNN?N y  32  N$e̙?"6@ NNN?N y  33  Ng?"6@ NNN?N)uI 30   Nk?"6@ NNN?N% 31   No?"6@ NNN?NI i  31   Nt?"6@ NNN?N   30   Nv?"6@ NNN?N   32   Nz?"6@ NNN?N ~ l  31  NL?"6@ NNN?N ] K  30  Np̙?"6@ NNN?N5 33  T ?"6@ NNN?Nf ZA 4-coloring f0 f  s *A &??, &B  TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDԔ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDԔ?"0@NNN?N CB @ ZDԔ?"0@NNN?N`B  ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?N&   N?"6@ NNN?N 5  32B   ZDԔ?"0@NNN?N  B !@ ZDԔ?"0@NNN?N? z ? B " ZDԔ?"0@NNN?N& B # ZDԔ?"0@NNN?N@ B $@ ZDԔ?"0@NNN?N  `B %@ ZDԔ?"0@NNN?N" Z B & ZDԔ?"0@NNN?Nj B ' ZDԔ?"0@NNN?N   ( TX ?"6@ NNN?Np W is acyclic   ) T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  * H< ?"6@ NNN?N) > 2 H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B ++P % %P'++$(    N?"6@ NNN?Ntb 31  Nd?"6@ NNN?N5 32  N4?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  N?"6@ NNN?N y  32  Nܴ̙?"6@ NNN?N y  33  N8?"6@ NNN?N)uI 30   N?"6@ NNN?N% 31   N?"6@ NNN?NI i  31   NL?"6@ NNN?N   30   N?"6@ NNN?N   32   N?"6@ NNN?N ~ l  31  N`?"6@ NNN?N ] K  30  N̙?"6@ NNN?N5 33  T ?"6@ NNN?Nf ZA 4-coloring f0 f  s *A &??, &B  TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDԔ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDԔ?"0@NNN?N CB @ ZDԔ?"0@NNN?N`B  ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?N&   NH?"6@ NNN?N 5  32B   ZDԔ?"0@NNN?N  B !@ ZDԔ?"0@NNN?N? z ? B " ZDԔ?"0@NNN?N& B # ZDԔ?"0@NNN?N@ B $@ ZDԔ?"0@NNN?N  `B %@ ZDԔ?"0@NNN?N" Z B & ZDԔ?"0@NNN?Nj B ' ZDԔ?"0@NNN?N   ( Tl ?"6@ NNN?Np W is acyclic   ) T  ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  * H ?"6@ NNN?N) > 2 B + T ?"6@ NNN?N/ K5,$  0 LMultiply each color by 3.H  0޽h ? fff̙5-___PPT10 .ne0 W2+E2D' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*+D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*++8+0++ +DP ))'/2)(    N?"6@ NNN?Ntb 31  ND?"6@ NNN?N5 32  N?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  Nh ?"6@ NNN?N y  32  Nl̙?"6@ NNN?N y  33  N8?"6@ NNN?N)uI 30   N?"6@ NNN?N% 31   N?"6@ NNN?NI i  31   NL?"6@ NNN?N   30   Np?"6@ NNN?N   32   N!?"6@ NNN?N ~ l  31  NX%?"6@ NNN?N ] K  30  N)̙?"6@ NNN?N5 33  T, ?"6@ NNN?Nf ZA 4-coloring f0 f  s *A &??, &B  TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDԔ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDԔ?"0@NNN?N CB @ ZDԔ?"0@NNN?N`B  ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?N&   N6?"6@ NNN?N 5  32B   ZDԔ?"0@NNN?N  B !@ ZDԔ?"0@NNN?N? z ? B " ZDԔ?"0@NNN?N& B # ZDԔ?"0@NNN?N@ B $@ ZDԔ?"0@NNN?N  `B %@ ZDԔ?"0@NNN?N" Z B & ZDԔ?"0@NNN?Nj B ' ZDԔ?"0@NNN?N   ( T< ?"6@ NNN?Np W is acyclic   + TH> ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  , H G ?"6@ NNN?N) > 2  . TH ?"6@ NNN?N/ K5 LMultiply each color by 3.# / NHM?"6@ NNN?N,$D 0 39# 0 N Q?"6@ NNN?N,$D 0 36# 1 NT?"6@ NNN?N v " ,$D 0 33# 2 NX?"6@ NNN?N  ,$D  0 36H  0޽h ? fff̙___PPT10.ne0 W2+6/D' = @B Dy' = @BA?%,( < +O%,( < +D' =%(%(D' =%(Df' =A@BB BB0B%()))D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-g6B fade*<3<*/D' =%(D' =%(Df' =A@BB BB0B%()))D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(Df' =A@BB BB0B%()))D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-g6B fade*<3<*1D' =%(D' =%(Df' =A@BB BB0B%()))D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-g6B fade*<3<*2++0+ ++0+  ++0+  ++0+ ++0+/ ++0+0 ++0+1 ++0+2 +$P ##0'*/I#(    Nl?"6@ NNN?Ntb 33  No?"6@ NNN?NV v 30  NPs?"6@ NNN?NL l  30  Nv?"6@ NNN?N y  36  Nz?"6@ NNN?N y  39  Nd}?"6@ NNN?N)uI 30   N?"6@ NNN?N% 33   N?"6@ NNN?NI i  33   Nx?"6@ NNN?N   30  NԊ?"6@ NNN?N ] K  30f  s *A &??, &B  TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDԔ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDԔ?"0@NNN?N CB @ ZDԔ?"0@NNN?N`B  ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?N&   ND?"6@ NNN?N 5  36B   ZDԔ?"0@NNN?N  B !@ ZDԔ?"0@NNN?N? z ? B " ZDԔ?"0@NNN?N& B # ZDԔ?"0@NNN?N@ B $@ ZDԔ?"0@NNN?N  `B %@ ZDԔ?"0@NNN?N" Z B & ZDԔ?"0@NNN?Nj B ' ZDԔ?"0@NNN?N   ( Th ?"6@ NNN?Np W is acyclic   ) T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  * H ?"6@ NNN?N) > 2  + T ?"6@ NNN?N/ K5 LMultiply each color by 3. , N?"6@ NNN?N 39 - N?"6@ NNN?N 36 . NP?"6@ NNN?N v "  33 / N ?"6@ NNN?N   36H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +;P $$`'+,j$(    N?"6@ NNN?Ntb 33  ND?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  N?"6@ NNN?N y  36  NX?"6@ NNN?N y  39  N?"6@ NNN?N)uI 30  N?"6@ NNN?N% 33   Nl?"6@ NNN?NI i  33   N?"6@ NNN?N   30   N$?"6@ NNN?N ] K  30f   s *A &??, &B  TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDԔ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDԔ?"0@NNN?N CB @ ZDԔ?"0@NNN?N`B  ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?N&   N?"6@ NNN?N 5  36B  ZDԔ?"0@NNN?N  B @ ZDԔ?"0@NNN?N? z ? B  ZDԔ?"0@NNN?N& B  ZDԔ?"0@NNN?N@ B  @ ZDԔ?"0@NNN?N  `B !@ ZDԔ?"0@NNN?N" Z B " ZDԔ?"0@NNN?Nj B # ZDԔ?"0@NNN?N   $ T ?"6@ NNN?Np W is acyclic   % T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  & Hx ?"6@ NNN?N) > 2  ' T< ?"6@ NNN?N/ K5 LMultiply each color by 3. ( N?"6@ NNN?N 39 ) Nx?"6@ NNN?N 36 * N?"6@ NNN?N v "  33 + ND?"6@ NNN?N   36 , TH ?"6@ NNN?N&, W!But this is not a (11,3)-coloring""H  0޽h ? fff̙___PPT10o.ne0 W2+D ' = @B D' = @BA?%,( < +O%,( < +D[' =%(%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*,D' =,54*3>Bfillcolor=@BPB<*,D' =,54*3>#Bstroke.color=@BPB<*,D' =1:B solid*a3>Bfill.type<*,D.' =%(D' =%(D~' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =,54*3>Bfillcolor=@BPB<*D' =,54*3>#Bstroke.color=@BPB<*D' =1:B solid*a3>Bfill.type<*D.' =%(D' =%(D~' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =,54*3>Bfillcolor=@BPB<*D' =,54*3>#Bstroke.color=@BPB<*D' =1:B solid*a3>Bfill.type<*D.' =%(D' =%(D~' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =,54*3>Bfillcolor=@BPB<*D' =,54*3>#Bstroke.color=@BPB<*D' =1:B solid*a3>Bfill.type<*+8+0+, +9P [(S(p'..'(    N`?"6@ NNN?Ntb 33  N?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  Nt?"6@ NNN?N y  36  N?"6@ NNN?N y  39  N,#?"6@ NNN?N)uI 30  N&?"6@ NNN?N% 33   N)?"6@ NNN?NI i  33   N@-?"6@ NNN?N   30   N0?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDԔ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDԔ?"0@NNN?N`B  ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   N 9?"6@ NNN?N 5  36B  ZDԔ?"0@NNN?N  B @ ZDԔ?"0@NNN?N? z ? B  ZDԔ?"0@NNN?N& B  ZDԔ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDԔ?"0@NNN?N   # T0? ?"6@ NNN?Np W is acyclic   $ T@ ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % HI ?"6@ NNN?N) > 2  & TlK ?"6@ NNN?N/ K5 LMultiply each color by 3. ' NO?"6@ NNN?N 39 ( NS?"6@ NNN?N 36 ) NX?"6@ NNN?N v "  33 * NZ?"6@ NNN?N   36 + T^ ?"6@ NNN?N&, W!But this is not a (11,3)-coloring""# , N_?"6@ NNN?N ~ l ,$@ 0 31# - N`g?"6@ NNN?N  ,$@ 0 31# . Nk?"6@ NNN?N9,$D 0 31H  0޽h ? fff̙___PPT10.ne0 W2+n9f3D{' = @B D6' = @BA?%,( < +O%,( < +Dm' =%(D' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-g6B fade*<3<*,D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-g6B fade*<3<*-D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-g6B fade*<3<*.++0+  ++0+  ++0+, ++0+- ++0+. +(P '''..O'(    N(|?"6@ NNN?Ntb 33  N?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  N<?"6@ NNN?N y  36  N?"6@ NNN?N y  39  N?"6@ NNN?N)uI 30  N?"6@ NNN?N% 33   N?"6@ NNN?NI i  33   Np?"6@ NNN?N   30   N̚?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   N<?"6@ NNN?N 5  36B  ZDԔ?"0@NNN?N  B @ ZDԔ?"0@NNN?N? z ? B  ZDԔ?"0@NNN?N& B  ZDԔ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T` ?"6@ NNN?Np W is acyclic   $ T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % Hس ?"6@ NNN?N) > 2  & T ?"6@ NNN?N/ K5 LMultiply each color by 3. ' N?"6@ NNN?N 39 ( Nؽ?"6@ NNN?N 36 ) NH?"6@ NNN?N v "  33 * N?"6@ NNN?N   36 + T ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N?"6@ NNN?N ~ l  31 - N\?"6@ NNN?N   31 . N,?"6@ NNN?N9 31H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '''..O'(    N?"6@ NNN?Ntb 33  NP?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  N?"6@ NNN?N y  36  Nd?"6@ NNN?N y  39  N?"6@ NNN?N)uI 30  N?"6@ NNN?N% 34   Nx?"6@ NNN?NI i  34   N?"6@ NNN?N   30   N0?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDԔ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   N?"6@ NNN?N 5  36B  ZDԔ?"0@NNN?N  B @ ZDԔ?"0@NNN?N? z ? B  ZDԔ?"0@NNN?N& B  ZDԔ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T ?"6@ NNN?Np W is acyclic   $ T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H ?"6@ NNN?N) > 2  & TL ?"6@ NNN?N/ K5 LMultiply each color by 3. ' N?"6@ NNN?N 39 ( N?"6@ NNN?N 36 ) N?"6@ NNN?N v "  34 * N"?"6@ NNN?N   36 + Th& ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N'?"6@ NNN?N ~ l  31 - N /?"6@ NNN?N   31 . N1?"6@ NNN?N9 31H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '''..O'(    N7?"6@ NNN?Ntb 33  N8;?"6@ NNN?NV v 30  N>?"6@ NNN?NL l  30  NB?"6@ NNN?N y  36  NF?"6@ NNN?N y  39  NxJ?"6@ NNN?N)uI 30  N(I?"6@ NNN?N% 34   NP?"6@ NNN?NI i  34   N|U?"6@ NNN?N   30   NS?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   N4`?"6@ NNN?N 5  36B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDԔ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # TXf ?"6@ NNN?Np W is acyclic   $ Tg ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % Hp ?"6@ NNN?N) > 2  & Tr ?"6@ NNN?N/ K5 LMultiply each color by 3. ' Nv?"6@ NNN?N 39 ( Nz?"6@ NNN?N 36 ) N@?"6@ NNN?N v "  34 * N?"6@ NNN?N   36 + T ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N ?"6@ NNN?N ~ l  31 - NT?"6@ NNN?N   31 . N$?"6@ NNN?N9 31H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '''..O'(    N?"6@ NNN?Ntb 33  NH?"6@ NNN?NV v 30  Nl?"6@ NNN?NL l  30  N|?"6@ NNN?N y  37  NT?"6@ NNN?N y  39  Nĩ?"6@ NNN?N)uI 30  N?"6@ NNN?N% 34   Nl?"6@ NNN?NI i  34   ND?"6@ NNN?N   30   N?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   Nd?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDԔ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T ?"6@ NNN?Np W is acyclic   $ T( ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H ?"6@ NNN?N) > 2  & T ?"6@ NNN?N/ K5 LMultiply each color by 3. ' N(?"6@ NNN?N 39 ( N?"6@ NNN?N 36 ) Np?"6@ NNN?N v "  34 * N@?"6@ NNN?N   37 + T ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N<?"6@ NNN?N ~ l  31 - N?"6@ NNN?N   31 . NT?"6@ NNN?N9 31H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '''..O'(    N?"6@ NNN?Ntb 33  Nx?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  ND?"6@ NNN?N y  37  N?"6@ NNN?N y  39  N4?"6@ NNN?N)uI 30  N ?"6@ NNN?N% 34   Nl?"6@ NNN?NI i  34   N?"6@ NNN?N   30   N?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   N?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDf3>?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T& ?"6@ NNN?Np W is acyclic   $ T' ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H0, ?"6@ NNN?N) > 2  & T1 ?"6@ NNN?N/ K5 LMultiply each color by 3. ' N(3?"6@ NNN?N 39 ( N8?"6@ NNN?N 36 ) N<?"6@ NNN?N v "  34 * N@?"6@ NNN?N   37 + TE ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N|I?"6@ NNN?N ~ l  31 - N$M?"6@ NNN?N   31 . NO?"6@ NNN?N9 31H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '''..P'(    NU?"6@ NNN?Ntb 33  NX?"6@ NNN?NV v 30  NL\?"6@ NNN?NL l  30  N_?"6@ NNN?N y  37  Nc?"6@ NNN?N  410  N`f?"6@ NNN?N)uI 30  Ni?"6@ NNN?N% 34   Nm?"6@ NNN?NI i  34   Ntp?"6@ NNN?N   30   Ns?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?N&   N@|?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # Td ?"6@ NNN?Np W is acyclic   $ T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H܌ ?"6@ NNN?N) > 2  & T?"0@NNN?N&   N?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T ?"6@ NNN?Np W is acyclic   $ TX ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H0 ?"6@ NNN?N) > 2  & T ?"6@ NNN?N/ K5 LMultiply each color by 3. ' NX?"6@ NNN?N 39 ( N0?"6@ NNN?N 36 ) N?"6@ NNN?N v "  34 * ND?"6@ NNN?N   37 + T ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N?"6@ NNN?N ~ l  31 - Nt ?"6@ NNN?N   31 . N?"6@ NNN?N9 31H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P ''(..P'(    N?"6@ NNN?Ntb 33  NP?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  N?"6@ NNN?N y  37  N"?"6@ NNN?N  410  NX&?"6@ NNN?N)uI 30  N4*?"6@ NNN?N% 34   N(?"6@ NNN?NI i  34   Nh1?"6@ NNN?N   30   N@5?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3Ԕ?"0@NNN?N`B  ZDf3Ԕ?"0@NNN?N&B  ZDf3>?"0@NNN?N&   N?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T? ?"6@ NNN?Np W is acyclic   $ TD ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % HF ?"6@ NNN?N) > 2  & TxL ?"6@ NNN?N/ K5 LMultiply each color by 3. ' NM?"6@ NNN?N 39 ( NU?"6@ NNN?N 36 ) NW?"6@ NNN?N v "  34 * N[?"6@ NNN?N   37 + T^ ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , NLc?"6@ NNN?N ~ l  31 - N$g?"6@ NNN?N   31 . Nj?"6@ NNN?N9 32H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '''..P'(    Npq?"6@ NNN?Ntb 33  NLu?"6@ NNN?NV v 30  Ns?"6@ NNN?NL l  30  N{?"6@ NNN?N y  37  NP?"6@ NNN?N  410  Nl~?"6@ NNN?N)uI 30  N?"6@ NNN?N% 34   N?"6@ NNN?NI i  34   N?"6@ NNN?N   30   N?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3>?"0@NNN?N`B  ZDf3>?"0@NNN?N&B  ZDf3>?"0@NNN?N&   N ?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T0 ?"6@ NNN?Np W is acyclic   $ TР ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H ?"6@ NNN?N) > 2  & Tl ?"6@ NNN?N/ K5 LMultiply each color by 3. ' NЯ?"6@ NNN?N 39 ( N?"6@ NNN?N 36 ) N?"6@ NNN?N v "  34 * N?"6@ NNN?N   37 + TS ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N?"6@ NNN?N ~ l  31 - N,?"6@ NNN?N   31 . N?"6@ NNN?N9 32H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P ''(..P'(    NC?"6@ NNN?Ntb 33  N(?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  N?"6@ NNN?N y  37  NH?"6@ NNN?N  410  N?"6@ NNN?N)uI 30  N?"6@ NNN?N% 35   N?"6@ NNN?NI i  35   N8?"6@ NNN?N   30   N?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3Ԕ?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3>?"0@NNN?N`B  ZDf3>?"0@NNN?N&B  ZDf3>?"0@NNN?N&   N?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3Ԕ?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T ?"6@ NNN?Np W is acyclic   $ T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H`  ?"6@ NNN?N) > 2  & T ?"6@ NNN?N/ K5 LMultiply each color by 3. ' N?"6@ NNN?N 39 ( N ?"6@ NNN?N 36 ) N?"6@ NNN?N v "  34 * N8?"6@ NNN?N   37 + T ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N!?"6@ NNN?N ~ l  31 - N%?"6@ NNN?N   31 . N)?"6@ NNN?N9 32H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P '' (..P'(    NH.?"6@ NNN?Ntb 33  Nh2?"6@ NNN?NV v 30  N5?"6@ NNN?NL l  30  N9?"6@ NNN?N y  37  N`=?"6@ NNN?N  410  N?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3>?"0@NNN?N`B  ZDf3>?"0@NNN?N&B  ZDf3>?"0@NNN?N&   NX?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3>?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # Td  ?"6@ NNN?Np W is acyclic   $ TXb ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H^ ?"6@ NNN?N) > 2  & Ti ?"6@ NNN?N/ K5 LMultiply each color by 3. ' Nn?"6@ NNN?N 39 ( Nq?"6@ NNN?N 36 ) Nu?"6@ NNN?N v "  34 * Ny?"6@ NNN?N   37 + Tpx ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N|?"6@ NNN?N ~ l  31 - NT?"6@ NNN?N   31 . N,?"6@ NNN?N9 32H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +(P ''0(..P'(    Nx?"6@ NNN?Ntb 33  N8?"6@ NNN?NV v 30  N?"6@ NNN?NL l  30  Nĕ?"6@ NNN?N y  38  N$?"6@ NNN?N  410  Nܟ?"6@ NNN?N)uI 30  N8?"6@ NNN?N% 35   N?"6@ NNN?NI i  35   N?"6@ NNN?N   30   NL?"6@ NNN?N ] K  30f   s *A &??, &B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3>?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3>?"0@NNN?N`B  ZDf3>?"0@NNN?N&B  ZDf3>?"0@NNN?N&   N?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3>?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   # T ?"6@ NNN?Np W is acyclic   $ T ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % HX ?"6@ NNN?N) > 2  & T ?"6@ NNN?N/ K5 LMultiply each color by 3. ' N?"6@ NNN?N 39 ( NX?"6@ NNN?N 36 ) N?"6@ NNN?N v "  34 * N?"6@ NNN?N   38 + T8 ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N?"6@ NNN?N ~ l  31 - N?"6@ NNN?N   31 . N?"6@ NNN?N9 32H  0޽h ? fff̙___PPT10i.ne0 W2+D=' = @B +.P ''`(-/'(    Nt?"6@ NNN?Ntb 33  N?"6@ NNN?NV v 30  N,?"6@ NNN?NL l  30  N?"6@ NNN?N y  38  N?"6@ NNN?N  410  N?"6@ NNN?N)uI 30  ND?"6@ NNN?N% 35   N?"6@ NNN?NI i  35   N4?"6@ NNN?N   30   N ?"6@ NNN?N ] K  30B   TD?"0@NNN?NG G B  ZDԔ?"0@NNN?N t`B @ ZDԔ?"0@NNN?NB  ZDԔ?"0@NNN?N B  ZDf3Ԕ?"0@NNN?N B @ ZDԔ?"0@NNN?N&B  ZDf3Ԕ?"0@NNN?Nna\ B  ZDf3>?"0@NNN?Nn IB  ZDԔ?"0@NNN?NW 2B  ZDf3Ԕ?"0@NNN?N CB @ ZDf3>?"0@NNN?N`B  ZDf3>?"0@NNN?N&B  ZDf3>?"0@NNN?N&   N?"6@ NNN?N 5  37B  ZDf3Ԕ?"0@NNN?N  B @ ZDf3Ԕ?"0@NNN?N? z ? B  ZDf3>?"0@NNN?N& B  ZDf3Ԕ?"0@NNN?N@ B @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B ! ZDԔ?"0@NNN?Nj B " ZDf3Ԕ?"0@NNN?N   $ T@ ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  % H ?"6@ NNN?N) > 2  & T" ?"6@ NNN?N/ K5 LMultiply each color by 3. ' Np#?"6@ NNN?N 39 ( N*?"6@ NNN?N 36 ) N-?"6@ NNN?N v "  34 * Nh/?"6@ NNN?N   38 + Tx2 ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" , N6?"6@ NNN?N ~ l  31 - N:?"6@ NNN?N   31 . N>?"6@ NNN?N9 32Q / TC ?"6@ NNN?NJ KP,$  0 [) Each vertex changed color at most twice**H  0޽h ? fff̙5-___PPT10 .ne0 W2+QQD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*/D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*/+8+0+/ +3P ))p(/0 @)(      NK?"6@ NNN?Ntb 33   NlO?"6@ NNN?NV v 30   NDS?"6@ NNN?NL l  30   NW?"6@ NNN?N y  38   NZ?"6@ NNN?N  410   N\^?"6@ NNN?N)uI 30   N4b?"6@ NNN?N% 35   Nf?"6@ NNN?NI i  35   Nti?"6@ NNN?N   30   NLm?"6@ NNN?N ] K  30B   TD?"0@NNN?NG G B   ZDԔ?"0@NNN?N t`B  @ ZDԔ?"0@NNN?NB   ZDԔ?"0@NNN?N B   ZDf3Ԕ?"0@NNN?N B  @ ZDԔ?"0@NNN?N&B   ZDf3Ԕ?"0@NNN?Nna\ B   ZDf3>?"0@NNN?Nn IB   ZDԔ?"0@NNN?NW 2B   ZDf3Ԕ?"0@NNN?N CB  @ ZDf3>?"0@NNN?N`B   ZDf3>?"0@NNN?N&B   ZDf3>?"0@NNN?N&    Nt?"6@ NNN?N 5  37B   ZDf3Ԕ?"0@NNN?N  B  @ ZDf3Ԕ?"0@NNN?N? z ? B   ZDf3>?"0@NNN?N& B   ZDf3Ԕ?"0@NNN?N@ B  @ ZDԔ?"0@NNN?N  `B  @ ZDԔ?"0@NNN?N" Z B   ZDԔ?"0@NNN?Nj B !  ZDf3Ԕ?"0@NNN?N   "  Tz ?"6@ NNN?NV \ Q How to get a (11, 3)-coloring  #  H ?"6@ NNN?N) > 2  $  T ?"6@ NNN?N/ K5 LMultiply each color by 3. %  ND?"6@ NNN?N 39 &  NT?"6@ NNN?N 36 '  N$?"6@ NNN?N v "  34 (  N?"6@ NNN?N   38 )  T ?"6@ NNN?N&, W!But this is not a (11,3)-coloring"" *  N?"6@ NNN?N ~ l  31 +  N?"6@ NNN?N   31 ,  Nܠ?"6@ NNN?N9 32 -  T ?"6@ NNN?NJ KP [) Each vertex changed color at most twice**B /  TD)?"0@NNN?N= ,$D 0b 0  T  ?"6@ NNN?Nq ,$ 0 lThe deleted edges are OK H   0޽h ? fff̙ { ___PPT10[ .ne0 W2+Q=D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*0 D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*0 D' =-g6B fade*<3<*0 D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*/ %(D' =-g6B fade*<3<*/ +8+0+0   +P )! *\(  \> \ T( ?"6@ NNN?N~ |JExercise: Find and prove a generalization of this lemma. KK \ N?"6@ NNN?N  5Lemma: If G has a 4-coloring f such that (1): is acyclic; (2): each directed path of contains at most 2 vertices of color 3, then 0f \ s *A <'??   <'f \ s *A ='?? 8  ='f \ s *A >'?? q 7  >'H \ 0޽h ? fff̙___PPT10i.NiJ9+D=' = @B + P   ( (    N?"6@ NNN?N -) c1G cubic, has a perfect matching, girth at least 422  H?"6@ NNN?NVPf  s *A &??  &  T ?"6@ NNN?N AFirst we prove:l     ,$D 0    T0 ?"6@ NNN?NF f  V$1: is acyclic;%%2  T| ?"6@ NNN?N   h6We shall construct a 4-edge coloring c of G so that 77h   c $A 3'??wV f  3'T   T ?"6@ NNN?N  X2: each directed path of contains at most two edges of color 3.YYh   c $A 4'?? ; 4'H  0޽h ? fff̙___PPT10n.e& +581DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+EP (%(  B   NDf3?"0@NNN?N  c B   NDf3?"0@NNN?N  ? B   NDf3?"0@NNN?N? T B  NDf3?"0@NNN?N< B  NDf3?"0@NNN?NgZB  NDf3?"0@NNN?N& B  NDf3?"0@NNN?NG fB  NDf3?"0@NNN?N)s B @ NDf3?"0@NNN?N  B  NDf3?"0@NNN?N% 9 B @ NDf3?"0@NNN?NB  NDf3?"0@NNN?N )f B  NDf3?"0@NNN?Nz3vB  NDf3?"0@NNN?N,B  NDf3?"0@NNN?Nn B  NDf3?"0@NNN?Ncg \ B  NDf3?"0@NNN?N B  NDf3?"0@NNN?N  B  NDf3?"0@NNN?N2  N  T ?"6@ NNN?NQR,$ 0 X&The M-edges will be colored by color 0''T  T  ?"6@ NNN?NIi,$ 0 ^,G has a perfect matching M (the brown edges)--"F n6    6n 2 ! N ?"6@ NNN?Nn2 " N ?"6@ NNN?N J S 2 # N ?"6@ NNN?N62 $ N ?"6@ NNN?N 2 % N ?"6@ NNN?N H  0޽h ? fff̙v0n0___PPT10N0.e+ZeD/' = @B Dm/' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D*' =%(D)' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* D' =0l9 BBBB*<3<* )?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* D' =0l9 BBBB*<3<* )?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* D' =0l9 BBBB*<3<* )?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*D' =0l9 BBBB*<3<*)?DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+ ++0+ +WP nf((  2  N ?"6@ NNN?Nn2  N ?"6@ NNN?N J S 2  N ?"6@ NNN?N62  N ?"6@ NNN?N 2  N ?"6@ NNN?N   T ?"6@ NNN?Np7 X&Delete the M-edges, we have a 2-factor''H  0޽h ? fff̙___PPT10i.e+D=' = @B +P   @)06 (  02 0 N ?"6@ NNN?Nn2 0 N ?"6@ NNN?N J S 2 0 Z\ ?"6@ NNN?N6 > 2 0 N ?"6@ NNN?N 2 0 N ?"6@ NNN?N  0 T  ?"6@ NNN?NwQi V$Color each even cycle by colors 1, 2%%  0 T ?"6@ NNN?NM4"m 31  0 T ?"6@ NNN?ND2 32  0 T ?"6@ NNN?N 31  0 T ?"6@ NNN?NG5  32 0 T ?"6@ NNN?NF^L f 31 0 T ?"6@ NNN?Nm7% 32 0 T,# ?"6@ NNN?N@v` 31 0 T& ?"6@ NNN?N 32H 0 0޽h ? fff̙___PPT10i.e+D=' = @B +=P nfP)4(  42 4 N ?"6@ NNN?Nn2 4 N ?"6@ NNN?N J S 2 4 Z,. ?"6@ NNN?N6 > 2 4 N ?"6@ NNN?N s 4 T41 ?"6@ NNN?N z,$  0 }KColor each odd cycle by colors 1, 2, except one edge is colored by color 3.LL 4 T2 ?"6@ NNN?NM4"m 31  4 T: ?"6@ NNN?ND2 32  4 T= ?"6@ NNN?N 31  4 T\< ?"6@ NNN?NG5  32  4 TD ?"6@ NNN?NF^L f 31  4 TG ?"6@ NNN?Nm7% 32 4 T\L ?"6@ NNN?N@v` 31 4 T,O ?"6@ NNN?N 32) 4 TS ?"6@ NNN?NL jXl ,$ 0 31) 4 TV ?"6@ NNN?N  ,$ 0 32) 4 TZ ?"6@ NNN?N  ,$ 0 31) 4 T/ ?"6@ NNN?N ) ,$ 0 32) 4 TLa ?"6@ NNN?N) #I,$ 0 31) 4 T$e ?"6@ NNN?Nm tb,$ 0 32) 4 Th ?"6@ NNN?N@ j X`,$  0 31) 4 T8m ?"6@ NNN?N ,$  0 32) 4 T$q ?"6@ NNN?Nc a O ,$  0 31) 4 Tu ?"6@ NNN?N ,$  0 32) 4 To ?"6@ NNN?N& F ,$  0 332 4 Z| ?"6@ NNN?N  > H 4 0޽h ? fff̙!!___PPT10!.e+@ MD' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*4Dh' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 ++0+4 +P y`)8(  82 8 N ?"6@ NNN?Nn2 8 N ?"6@ NNN?N J S 2 8 Z ?"6@ NNN?N6 > 2 8 N ?"6@ NNN?N  8 T ?"6@ NNN?N   ,$D  0 \Such a 4-edge coloring of G is called a VALID edge coloring of G (w.r.t. the matching M). 2](0D 8 T| ?"6@ NNN?NM4"m 31 8 Tĭ ?"6@ NNN?ND2 32  8 T ?"6@ NNN?N 31  8 Tt ?"6@ NNN?NG5  32  8 T ?"6@ NNN?NF^L f 31  8 T ?"6@ NNN?Nm7% 32  8 T ?"6@ NNN?N@v` 31 8 Td ?"6@ NNN?N 32 8 T ?"6@ NNN?NL jXl  31 8 T ?"6@ NNN?N   32 8 T| ?"6@ NNN?N   31 8 TT ?"6@ NNN?N )  32 8 T ?"6@ NNN?N) #I 31 8 T ?"6@ NNN?Nm tb 32 8 T< ?"6@ NNN?N@ j X` 31 8 T@ ?"6@ NNN?N  32 8 T ?"6@ NNN?Nc a O  31 8 T ?"6@ NNN?N  32 8 TT ?"6@ NNN?N& F  332 8 Zx ?"6@ NNN?N  > H 8 0޽h ? fff̙ ___PPT10.e+(z3D' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*8+8+0+8 +P  ) ,e(  , , T ?"6@ NNN?N&5F [)Given a valid 4-edge coloring c of G, is **` , c $A 5'??c 5'* , TD ?"6@ NNN?N  h 1: acyclic ?o , T\ ?"6@ NNN?N  Y2: every directed path of contains at mots 2 edges of color 3 ? ZZ?` , c $A 6'?? l 6'4  , T  ?"6@ NNN?Nz { ,$ 0 > It depends.  S  , H ?"6@ NNN?N? #E,$  0 i7We shall modify the coloring c so that 1 and 2 hold.88H , 0޽h ? fff̙___PPT10n.h@+ UD' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* ,D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* ,+p+0+ , ++0+ , +y#P >6p)<(  < < T ?"6@ NNN?NQ  < % < T ?"6@ NNN?N c11 and 2 can be combined into a single requirement22K < N$?"6@ NNN?NT ]1&2: Every directed walk of contains at most two edges of color 3.^^f < s *A 7'?? l# 7'Z  < T" ?"6@ NNN?NT%,$  0 d2Only odd cycles of G-M contains edges of color 3.33 l   < ,$D 02 < Z( ?"6@ NNN?Ns B >   < T+ ?"6@ NNN?N )  33  < T  ?"6@ NNN?N @ ` 31  < Tx' ?"6@ NNN?N3 !  32 < T4 ?"6@ NNN?N^PL p 31 < T8 ?"6@ NNN?N   32 < TL; ?"6@ NNN?N y )  31 < T> ?"6@ NNN?N  32B < ND?"0@NNN?NZ  B < ND?"0@NNN?N`  B < ND?"0@NNN?Ng 6 B < ND?"0@NNN?N B B <B ND?"0@NNN?N0 ] B < ND?"0@NNN?Nz  B < ND?"0@NNN?NvBY < TD ?"6@ NNN?N  ,$  0 c!Red edges are colored by color 0&" H < 0޽h ? fff̙  ___PPT10 .FiUa+CD ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* <%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* <D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* <D ' =%(D' =%(D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*<+p+0+ < ++0+< +l"P !{!)+0@!(  @2 @ H̙?"6@ NNN?NCw2 @ H̙?"6@ NNN?N B '@ TD̙>?"0@NNN?Nd mB %@ TD̙>?"0@NNN?N B &@@ TD̙>?"0@NNN?N6 t ) @ ZdS ?"6@ NNN?N  33 @ ZV ?"6@ NNN?Nu 31 @ ZZ ?"6@ NNN?N | <  32  @ Zx] ?"6@ NNN?N  31  @ Z` ?"6@ NNN?N)I 32  @ Z0d ?"6@ NNN?No 9 31  @ Zg ?"6@ NNN?N  32B  @ TD?"0@NNN?N""B @ TD?"0@NNN?NC 7j B @ TD?"0@NNN?N CB @ TD?"0@NNN?N@ h B @@ TD?"0@NNN?N B @ TD?"0@NNN?NSB @ TD?"0@NNN?N~T2 @ H ?"6@ NNN?N @ Tm ?"6@ NNN?N0 Z(The line graph L(G) around the cycle is ))2 @ H̙?"6@ NNN?NC2 @ H̙?"6@ NNN?Nf2 @ H̙?"6@ NNN?N)d 2 @ H̙?"6@ NNN?Nmw 2 @ H̙?"6@ NNN?Np  2 @ H̙?"6@ NNN?N] 2 @ H̙?"6@ NNN?N ]2  @ H̙?"6@ NNN?Np~ 2 !@ H̙?"6@ NNN?NP2 "@ H̙?"6@ NNN?N2 #@ H̙?"6@ NNN?Ni m 2 $@ H̙?"6@ NNN?N-pB (@ TD̙>?"0@NNN?N@B )@ TD̙>?"0@NNN?NZPB *@@ TD̙>?"0@NNN?N fB +@ TD̙>?"0@NNN?N/ B ,@ TD̙>?"0@NNN?N] mB -@ TD̙>?"0@NNN?N w B .@ TD̙>?"0@NNN?N B /@@ TD̙>?"0@NNN?N92B 0@ TD̙>?"0@NNN?N pH @ 0޽h ? fff̙___PPT10i.GiP̞}+D=' = @B +/P )3D)(  D D Z{ ?"6@ NNN?N= + 33 D Z( ?"6@ NNN?N){iI 31 D Z ?"6@ NNN?N2#  R 32 D Z ?"6@ NNN?N9 Y 31 D Z< ?"6@ NNN?N7% 32 D Z ?"6@ NNN?N*   31 D Z ?"6@ NNN?Na O  32B D TD?"0@NNN?N]B D TD?"0@NNN?N B D TD?"0@NNN?N sB  D TD?"0@NNN?N+ P B !D@ TD?"0@NNN?N sB "D TD?"0@NNN?Nc-B #D TD?"0@NNN?N_g=+2 $D N ?"6@ NNN?N/=p %D T< ?"6@ NNN?NZH? 30 &D T ?"6@ NNN?N  30 'D T ?"6@ NNN?N  30 (D TPo ?"6@ NNN?N 30 )D T ?"6@ NNN?NPp 30 *D T ?"6@ NNN?N  30 +D T ?"6@ NNN?NG5 30I ,D TH ?"6@ NNN?N J  UConsider directed walks of starting and ending with edges of color 3. VV` -D c $A 8'?? ~   8'X .D T ?"6@ NNN?N},$  0 b0Which M-edges will be contained in such a walk ?11 /D N?"6@ NNN?NE ,$D  0B 0D ZD̙>?"0@NNN?NC C,$D  0B 1D ZD̙>?"0@NNN?NC C,$D  0B 2D EF"Q&UVWԔ? ~~]%%it  t-p^t %;)^-;^0w258c9:G;a<G<<#=|0>>\AvBHD4 E Gx _H7vI$JiHKG_K%`T%`T?S C1REPGwOIMKKaMvIOFiPvCQ@R'=(=)L=+==,j=^-=u.J>/>/e?/@G/RA.AG.OB-B@E`@`@`@0*`T%0*`T%&R;LockC3  ,$D 0H D 0޽h ? fff̙___PPT10.KiPwx+&D9' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.D%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*.DD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*.DD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/D%(D' =-s6Bwipe(down)*<3<*/DD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0D%(D' =-s6Bwipe(left)*<3<*0DD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1D%(D' =-s6Bwipe(left)*<3<*1DD' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*2D%(D' =-g6B fade*<3<*2D+8+0+.D +IP `X)!H(  H H TH ?"6@ NNN?N= + 33 H T ?"6@ NNN?N){iI 31 H T ?"6@ NNN?N2#  R 32 H T\ ?"6@ NNN?N9 Y 31 H T ?"6@ NNN?N7% 32 H T ?"6@ NNN?N*   31 H Tp ?"6@ NNN?Na O  32B  H ND?"0@NNN?N]B  H ND?"0@NNN?N B  H ND?"0@NNN?N sB  H ND?"0@NNN?N+ P B  H@ ND?"0@NNN?N sB H ND?"0@NNN?Nc-B H ND?"0@NNN?N_g=+2 H N ?"6@ NNN?N/=p H T ?"6@ NNN?NZH? 30 H TD ?"6@ NNN?N  30 H T ?"6@ NNN?N  30 H T4 ?"6@ NNN?N 30 H T ?"6@ NNN?NPp 30 H T ?"6@ NNN?N  30 H T ?"6@ NNN?NG5 30 H N?"6@ NNN?NE   H N?"6@ NNN?N  !H N?"6@ NNN?Np % H H 0޽h ? fff̙___PPT10i.KiPwx+D=' = @B +'P 80)!L(  L L Tl ?"6@ NNN?N  33 L T ?"6@ NNN?N){iI 31 L T ?"6@ NNN?N2#  R 32 L TD ?"6@ NNN?N9 Y 31 L Tp ?"6@ NNN?N7% 32 L T  ?"6@ NNN?N*   31 L T ?"6@ NNN?Na O  32B  L ND?"0@NNN?N]B  L ND?"0@NNN?N B  L ND?"0@NNN?N sB  L ND?"0@NNN?N+ P B  L@ ND?"0@NNN?N sB L ND?"0@NNN?Nc-B L ND?"0@NNN?N_g=+2 L N ?"6@ NNN?N/=p L T ?"6@ NNN?NZH? 30 L T ?"6@ NNN?N  30 L T\ ?"6@ NNN?N  30 L T ?"6@ NNN?N 30 L T # ?"6@ NNN?NPp 30 L T& ?"6@ NNN?N  30 L T! ?"6@ NNN?NG5 30 L N?"6@ NNN?NE  L N?"6@ NNN?N  L N?"6@ NNN?Np % B L ZD̙>?"0@NNN?Nd s,$D  0B L@ ZD̙>?"0@NNN?N6 F,$D 0B  L ZD̙>?"0@NNN?NcCC,$D  0B !L ZD̙>?"0@NNN?NZ m,$D  0H L 0޽h ? fff̙  ___PPT10 .KiPwx+LpD ' = @B D{ ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bwipe(up)*<3<*LD' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-u6Bwipe(right)*<3<*LD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-s6Bwipe(down)*<3<* LD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!L%(D' =-s6Bwipe(left)*<3<*!L+P )#P0(  P P T84 ?"6@ NNN?N  33 P T7 ?"6@ NNN?N){iI 31 P T: ?"6@ NNN?N2#  R 32 P TL> ?"6@ NNN?N9 Y 31 P TA ?"6@ NNN?N7% 32 P TE ?"6@ NNN?N*   31 P T`H ?"6@ NNN?Na O  32B  P ND?"0@NNN?N]B  P ND?"0@NNN?N B  P ND?"0@NNN?N sB  P ND?"0@NNN?N+ P B  P@ ND?"0@NNN?N sB P ND?"0@NNN?Nc-B P ND?"0@NNN?N_g=+2 P N ?"6@ NNN?N/=p P TpN ?"6@ NNN?NZH? 30 P TxS ?"6@ NNN?N  30 P TV ?"6@ NNN?N  30 P TP ?"6@ NNN?N 30 P TY ?"6@ NNN?NPp 30 P Tx` ?"6@ NNN?N  30 P Td ?"6@ NNN?NG5 30l #P TLg ?"6@ NNN?N  xFor each odd cycle, at most 4 M-edges incident to it can be used in a directed walk which contains 3 edges of color 3. yyH P 0޽h ? fff̙___PPT10i.KiPwx+D=' = @B +$P *T0(  T T Tn ?"6@ NNN?N  33 T Tr ?"6@ NNN?N){iI 31 T Tv ?"6@ NNN?N2#  R 32 T Tu ?"6@ NNN?N9 Y 31 T T0} ?"6@ NNN?N7% 32 T Tl ?"6@ NNN?N*   31 T T  ?"6@ NNN?Na O  32B  T ZDԔ?"0@NNN?N],$D  0B  T ND?"0@NNN?N B  T ZDԔ?"0@NNN?N s,$D  0B  T ND?"0@NNN?N+ P B  T@ ZDԔ?"0@NNN?N s,$D  0B T ZDԔ?"0@NNN?Nc-,$D  0B T ND?"0@NNN?N_g=+2 T N ?"6@ NNN?N/=p T T ?"6@ NNN?NZH? 30 T T@ ?"6@ NNN?N  30 T Tď ?"6@ NNN?N  30 T T ?"6@ NNN?N 30 T Tؙ ?"6@ NNN?NPp 30 T T@ ?"6@ NNN?N  30 T T4 ?"6@ NNN?NG5 30l T TX ?"6@ NNN?N  xFor each odd cycle, at most 4 M-edges incident to it can be used in a directed walk which contains 3 edges of color 3. yyH T 0޽h ? fff̙  ___PPT10 .KiPwx+pRD ' = @B Du ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-o6Bwipe(up)*<3<*TD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =-o6Bwipe(up)*<3<* TD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =-s6Bwipe(down)*<3<* TD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =-s6Bwipe(left)*<3<* T+8P !!*%&Xs!(  X X TH ?"6@ NNN?N  33 X T ?"6@ NNN?N){iI 31 X T ?"6@ NNN?N2#  R 32 X T\ ?"6@ NNN?N9 Y 31 X T ?"6@ NNN?N7% 32 X T ?"6@ NNN?N*   31 X Tp ?"6@ NNN?Na O  32B  X ZDԔ?"0@NNN?N]B  X ND?"0@NNN?N B  X ZDԔ?"0@NNN?N sB  X ND?"0@NNN?N+ P B  X@ ZDԔ?"0@NNN?N sB X ZDԔ?"0@NNN?Nc-B X ND?"0@NNN?N_g=+2 X N ?"6@ NNN?N/=p X T ?"6@ NNN?NZH? 30 X T ?"6@ NNN?N  30 X Tp ?"6@ NNN?N  30 X T ?"6@ NNN?N 30 X T( ?"6@ NNN?NPp 30 X T ?"6@ NNN?N  30 X T ?"6@ NNN?NG5 30l X Tl ?"6@ NNN?N  xFor each odd cycle, at most 4 M-edges incident to it can be used in a directed walk which contains 3 edges of color 3. yy X T0 ?"6@ NNN?N +  LCompletely blocked M-edgesB X TD?"0@NNN?N B X TD?"0@NNN?NC B X TD?"0@NNN?N S A X TT ?"6@ NNN?Ni ,$ 0 KOne direction is blocked B X TD?"0@NNN?N ],$@  0B  X TD?"0@NNN?N c,$@  0B !X TD?"0@NNN?NS@ ,$@  0B "X TD?"0@NNN?N< 6@,$D  0B #X TD?"0@NNN?Nj,$@  0B $X TD?"0@NNN?N ],$@  0B %X@ TD?"0@NNN?NMJ s,$@   0B &X@ TD?"0@NNN?Nt,$D  0H X 0޽h ? fff̙qi___PPT10I.KiPwx+ =D' = @B D' = @BA?%,( < +O%,( < +D- ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-o6Bwipe(up)*<3<*XD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-o6Bwipe(up)*<3<* XD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!X%(D' =-o6Bwipe(up)*<3<*!XD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"X%(D' =-o6Bwipe(up)*<3<*"XD ' =%(DJ ' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#X%(D' =-s6Bwipe(down)*<3<*#XD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$X%(D' =-s6Bwipe(down)*<3<*$XD3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%X%(D' =-o6Bwipe(up)*<3<*%XD9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&X%(D' =-u6Bwipe(right)*<3<*&X+8+0+X +G,P @*'`D(  ` ` T ?"6@ NNN?N  33 ` T ?"6@ NNN?N){iI 31 ` T| ?"6@ NNN?N2#  R 32 ` T ?"6@ NNN?N9 Y 31 ` T  ?"6@ NNN?N7% 32 ` Tl ?"6@ NNN?N*   31 ` T ?"6@ NNN?Na O  32B  ` ZDԔ?"0@NNN?N]B  ` ND?"0@NNN?N B  ` ZDԔ?"0@NNN?N sB  ` ND?"0@NNN?N+ P B  `@ ZDԔ?"0@NNN?N sB ` ZDԔ?"0@NNN?Nc-B ` ND?"0@NNN?N_g=+2 ` N ?"6@ NNN?N/=p ` T ?"6@ NNN?NZH? 30 ` Tl ?"6@ NNN?N  30 ` T# ?"6@ NNN?N  30 ` T  ?"6@ NNN?N 30 ` T* ?"6@ NNN?NPp 30 ` T- ?"6@ NNN?N  30 ` T 2 ?"6@ NNN?NG5 302 "` N ?"6@ NNN?N,-m. #` T5 ?"6@ NNN?N,$ 0 8 not 3. $` T: ?"6@ NNN?Ny",$ 0 8 not 3N %` T= ?"6@ NNN?N ^? ,$  0 X&This M-edge becomes completely blocked''B &` TD?"0@NNN?Ns ,$D  0B '` ND?"0@NNN?N-j,$D 0H ` 0޽h ? fff̙;3___PPT10.KiPwx+GD?' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#`%(D3' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%`%(D' =-s6Bwipe(down)*<3<*%`D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&`%(D' =-s6Bwipe(down)*<3<*&`Dd' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*`D' =1:Bhidden*o3>+B#style.visibility<*`%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*'`%(D' =-g6B fade*<3<*'`++0+#` ++0+$` ++0+%` +P `*hI(  h h TM ?"6@ NNN?N  33 h TP ?"6@ NNN?N){iI 31 h TPT ?"6@ NNN?N2#  R 32 h TW ?"6@ NNN?N9 Y 31 h T[ ?"6@ NNN?N7% 32 h T,^ ?"6@ NNN?N*   31 h T_ ?"6@ NNN?Na O  32B  h ZDԔ?"0@NNN?N]B  h ND?"0@NNN?N B  h TDg ?"0@NNN?N sB  h ND?"0@NNN?N+ P B  h@ TDg ?"0@NNN?N sB h ZDԔ?"0@NNN?Nc-B h ND?"0@NNN?N_g=+2 h N ?"6@ NNN?N/=p h Th ?"6@ NNN?NZH? 30 h TXl ?"6@ NNN?N  30 h T k ?"6@ NNN?N  30 h Tr ?"6@ NNN?N 30 h Tv ?"6@ NNN?NPp 30 h Tu ?"6@ NNN?N  30 h T} ?"6@ NNN?NG5 30a h T@ ?"6@ NNN?N  mBy considering the coloring of the other odd cycles, more M-edges incident to C become completely blocked. nnH h 0޽h ? fff̙___PPT10i.KiPwx+D=' = @B +Q&P )!P*$d(  d d T ?"6@ NNN?NF f 33 d T ?"6@ NNN?Nq_ 31 d T̉ ?"6@ NNN?N   32 d Tx ?"6@ NNN?N  31 d TH ?"6@ NNN?N-9 32 d Td ?"6@ NNN?N   31 d T ?"6@ NNN?N&W E F 32B  d ZDԔ?"0@NNN?NSB  d ND?"0@NNN?NI pB  d TDg ?"0@NNN?N }B  d ND?"0@NNN?Nz F XB  d@ TDg ?"0@NNN?N B d ZDԔ?"0@NNN?N#wB d ND?"0@NNN?N]32 d N ?"6@ NNN?N3 d TH ?"6@ NNN?NP> 30 d T ?"6@ NNN?N y 6 30 d T4 ?"6@ NNN?Nq  30 d T ?"6@ NNN?NM m 30 d TD ?"6@ NNN?N 30 d T0 ?"6@ NNN?N < 30 d T ?"6@ NNN?NP=+p 30 d Td ?"6@ NNN?Nj < Input of C B d TD?"0@NNN?N }  d Tl ?"6@ NNN?N&F = Output of C B "d@ TD?"0@NNN?N6 #d Tx ?"6@ NNN?Np\ ,$  0 ~Lemma: There is a valid 4-edge coloring c such that no odd cycle of G-M has both input and output.  $d T ?"6@ NNN?N ,$  0 vThis implies that no directed walk contains 3 edges of color 3, because each odd cycle has only one edge of color 3. wwH d 0޽h ? fff̙  ___PPT10 .KiPwx+BD ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#d%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*#dD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*#dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$d%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$dD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*$d+p+0+#d ++0+$d +P p*lN(  l l T ̙?"6@ NNN?N  33 l T ?"6@ NNN?N){iI 31 l Tx ?"6@ NNN?N2#  R 32 l T ?"6@ NNN?N9 Y 31 l T ?"6@ NNN?N7% 32 l T ?"6@ NNN?N*   31 l Th ?"6@ NNN?Na O  32B  l ZDԔ?"0@NNN?N]B  l ND?"0@NNN?N B  l TDg ?"0@NNN?N sB  l ND?"0@NNN?N+ P B  l@ TDg ?"0@NNN?N sB l ZDԔ?"0@NNN?Nc-B l ND?"0@NNN?N_g=+2 l N ?"6@ NNN?N/=p l T ?"6@ NNN?NZH? 30 l T ?"6@ NNN?N  30 l T ?"6@ NNN?N  30 l T ?"6@ NNN?N 30 l T ?"6@ NNN?NPp 30 l TD ?"6@ NNN?N  30 l Tp ?"6@ NNN?NG5 30 l H  ?"6@ NNN?N > 2 2 l N ?"6@ NNN?NpC2 l N ?"6@ NNN?NFC l T ̙?"6@ NNN?Nwk 33 l T ̙?"6@ NNN?Nj^ 33H l 0޽h ? fff̙___PPT10i.KiPwx+D=' = @B + P   *p] (  pi p T8 ?"6@ NNN?Ni U  [ number of not completely blocked M-edges which are chords of some odd cycles C of G-M\\[P p H ?"6@ NNN?NSWmY N number of not completely blocked M-edges connecting two distinct odd cycles.OON` p c $A A'??  A'B  p TD̙?"0@NNN?N   p T ?"6@ NNN?N2 R 33 p H$ ?"6@ NNN?N 8  31B p TD̙?"0@NNN?N z  o2 p N ?"6@ NNN?N 2 p N ?"6@ NNN?N" ,` p c $A D'??   D'2 p N ?"6@ NNN?Nm & p T( ?"6@ NNN?N9Y 33 p T+ ?"6@ NNN?N W E  31H p 0޽h ? fff̙___PPT10i.Ui[+D=' = @B +P *t5(  t t Tl1 ?"6@ NNN?N*-y  Lemma: If c is a valid 4-edge coloring of G such that is minimum, then no odd cycles C has both inputs and outputs. ` t c $A E'??"  E'H t 0޽h ? fff̙___PPT10i.Vi f+D=' = @B +P `X*x(  x x T8 ?"6@ NNN?Nu/ \*Assume c is a valid 4-edge coloring of G. ++2 x N ?"6@ NNN?NC,$@ 0| x T= ?"6@ NNN?N)/,$ 0 : is an odd cycle which contains both inputs and outputs. ;;: x T@B ?"6@ NNN?N * ,$  0 bWe shall find another valid coloring c such that22 x c $A! F'??m Ge 8 F'$D  0H x 0޽h ? fff̙  ___PPT10 .Wi`b+jʁD$ ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*xD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*xD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*xD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*x+p+0+x ++0+x +` P *|<(  |2 | N ?"6@ NNN?N- F | T|P ?"6@ NNN?N  0First we consider the case that C is chordless. 11% Z | TU ?"6@ NNN?N ; ,$ 0 dUncolor the edges of C.  | ThZ ?"6@ NNN?Nm Y,$ 0 pEach M-edge incident to C has at least one direction blocked, because it is incident to another colored cycle.qqH | 0޽h ? fff̙\T___PPT104.Xi պ+nD' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(+p+0+| ++0+| +P b Z +  (  2  N ?"6@ NNN?N- B  TD?"0@NNN?N&/B @ TD?"0@NNN?N< B @ TD?"0@NNN?N B   TD?"0@NNN?NZB   TD?"0@NNN?NTB   TD?"0@NNN?N] pB   TD?"0@NNN?N w pB   TD?"0@NNN?N 0  Tdh ?"6@ NNN?N   Each M-edge has at least one blocked direction (under the condition that C is uncolored). For simplicity, for each M-edge incident to C, we draw exactly one direction blocked. q  Tm ?"6@ NNN?NIO,$ 0 {IBy assumption, at least one edge is an input and one edge is an output. JJH  0޽h ? fff̙___PPT10.Xi պ+IDO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +.P +_(  2  N ?"6@ NNN?N- B  TD?"0@NNN?N&/B @ TD?"0@NNN?N< B @ TD?"0@NNN?N B  TD?"0@NNN?NZB  TD?"0@NNN?NTB  TD?"0@NNN?N] pB   TD?"0@NNN?N w pB   TD?"0@NNN?N 0I   Tz ?"6@ NNN?Nj pp  UAs C is odd, there are two consecutive M-edges that have a common blocked direction. VV    0e0e    BCDE|F6 A@  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<WrNEA>3;]0d-)"xEq|~U<@          `s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab` `,$D  0B  TD?"0@NNN?NM& z ,$D  0j  T, ?"6@ NNN?Nm (s,$ 0 tBThe next two consecutive M-edges have opposite blocked directions.CC    R0e0e    BC.DEF^ A@  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||==+.v~tpfbZOwOwHAI+!Sk%y   "ex@2F\.!D&\4nAbboywc.O.&.?+.,0@                    `s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abR,$D  0B  TD?"0@NNN?N~O ,$D  0H  0޽h ? fff̙___PPT10f.Xi պ+F D' = @B D' = @BA?%,( < +O%,( < +D~' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D~' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*+8+0+ +S.P ph +(  2  N ?"6@ NNN?N- B  TD?"0@NNN?N&/B @ TD?"0@NNN?N< B @ TD?"0@NNN?N B  TD?"0@NNN?NZB  TD?"0@NNN?NTB  TD?"0@NNN?N] pB   TD?"0@NNN?N w pB   TD?"0@NNN?N 0    0e0e    BCDE|F6 @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<WrNEA>3;]0d-)"xEq|~U<@          `s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab` `    X0e0e    BC.DEF^ @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||==+.v~tpfbZOwOwHAI+!Sk%y   "ex@2F\.!D&\4nAbboywc.O.&.?+.,0@                    `s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abR  T ?"6@ NNN?Nz *H  CColor C as above.)  T$ ?"6@ NNN?N r ,$  0 31)  T ?"6@ NNN?N,$  0 32)  TT ?"6@ NNN?NF7%f,$  0 33H  0޽h ? fff̙___PPT10c.Xi պ+,D' = @B DJ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ +/P !!0+!(  2  N ?"6@ NNN?N- B  TD?"0@NNN?N&/B @ TD?"0@NNN?N< B @ TD?"0@NNN?N B  TD?"0@NNN?NZB  TD?"0@NNN?NTB  TD?"0@NNN?N] pB   TD?"0@NNN?N w pB   TD?"0@NNN?N 0    0e0e    BCDE|F6 @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<WrNEA>3;]0d-)"xEq|~U<@          `s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab` `     X0e0e    BC.DEF^ @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||==+.v~tpfbZOwOwHAI+!Sk%y   "ex@2F\.!D&\4nAbboywc.O.&.?+.,0@                    `s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abR   T ?"6@ NNN?Nz *H  CColor C as above.  T4 ?"6@ NNN?N r  31  T  ?"6@ NNN?N 32  T ?"6@ NNN?NF7%f 33B  TD?"0@NNN?N  ,$D  0B  TD?"0@NNN?N/,$D  0B  TD?"0@NNN?N3j,,$D  0H  T ?"6@ NNN?N W,$ 0 R We have more blocked directions.!!H  0޽h ? fff̙  ___PPT10 .Xi պ+D< ' = @B D ' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*+8+0+ +@P $$@+$(  2  N ?"6@ NNN?N- B  TD?"0@NNN?N&/B @ TD?"0@NNN?N< B @ ND?"0@NNN?N ,$D 0B  TD?"0@NNN?NZB  TD?"0@NNN?NTB  ND?"0@NNN?N] p,$@ 0B   ND?"0@NNN?N w p,$@ 0B   ND?"0@NNN?N 0,$@ 0    0e0e    BCDE|F6 @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<WrNEA>3;]0d-)"xEq|~U<@          `s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab` `     X0e0e    BC.DEF^ @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||==+.v~tpfbZOwOwHAI+!Sk%y   "ex@2F\.!D&\4nAbboywc.O.&.?+.,0@                    `s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abR  T, ?"6@ NNN?N r  31  T ?"6@ NNN?N 32  T ?"6@ NNN?NF7%f 33B  TD?"0@NNN?N  B  TD?"0@NNN?N/B  TD?"0@NNN?N3j,b  T< ?"6@ NNN?N\ xb ,$ 0 l:At most one M-edge incident to C is not completely blocked;;`  c $A' G'?? |  G'r  T ?"6@ NNN?N X,$  0 |JOriginally at least two M-edges incident to C are not completely blocked. KK  s *A& H'??3c 8 H'$D   0X  T ?"6@ NNN?N"<,$ 0 b0M-edges not incident to C remain the same status11H  0޽h ? fff̙___PPT10.Xi պ+0D' = @B Dm' = @BA?%,( < +O%,( < +D( ' =%(%(D ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bdiamond(in)*<3<* D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bdiamond(in)*<3<* D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ + P umP+(    T ?"6@ NNN?N=] Z(Next we consider the case C has a chord.))2  N ?"6@ NNN?NF B  ND̙?"0@NNN?N 6 7  TD ?"6@ NNN?Nh Q] ,$ 0 ADetails omitted$H  0޽h ? fff̙  ___PPT10._i4*+,DW' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+p+0+ ++0+ +P p+ u(     T@  ?"6@ NNN?N& F KUp to now we have proved   N?"6@ NNN?N% c1G cubic, has a perfect matching, girth at least 422  H?"6@ NNN?NR AL f  s *A I'??.  I'o   N?"6@ NNN?N  Corollary: If G is 2-edge connected, has maximum degree at most 3, and girth at least 4, then f   s *A Q'??   Q'H  0޽h ? fff̙___PPT10i.`iR+D=' = @B +!P   ,  (  (  Tt ?"6@ NNN?NK f4Proof. If G is cubic, then G has a perfect matching.55  T ?"6@ NNN?Ns* = Otherwise,  2   N?"6@ NNN?N ,$@ 0)   T ?"6@ NNN?N ,$ 0 3G2   N?"6@ NNN?N  ,$@ 0)   T ?"6@ NNN?Na j ,$ 0 3GB  ND?"0@NNN?N,,$@ 0B  ND?"0@NNN?N - ,$@ 0,  T( ?"6@ NNN?N ,$ 0 6G   T. ?"6@ NNN?N c,$   0 G is cubic, either 2-edge connected, or contains one cut edge. In any case G has a perfect matchingggH  0޽h ? fff̙nf___PPT10F.gi0iN+aD:' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*++0+  ++0+  ++0+ ++0+ +. P TL+(     ThB ?"6@ NNN?N& F KUp to now we have proved   NF?"6@ NNN?N% c1G cubic, has a perfect matching, girth at least 422  H?"6@ NNN?NR AL f  s *A P'??.  P'L  TK ?"6@ NNN?N@ F XNext we omit the condition that G has girth at least 4, cubic, has a perfect matching.YYH  0޽h ? fff̙z___PPT10Z.`iR+D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*+8+0+ +P + v(  .  T`U ?"6@ NNN?N=C l:We prove it by induction on the number of vertices that ;;[  NY?"6@ NNN?N1  KG has maximum degree 3, and does not contain as subgraphsLLB   H?"6@ NNN?NR AL f  s *A J'??% . -  J'`  c $A# K'??a  K'   T_ ?"6@ NNN?N  W%We may assume G is 2-edge connected. &&H  0޽h ? fff̙___PPT10i.aiU+D=' = @B +eP |t+ (    Ne?"6@ NNN?N==C c1G has maximum degree 3, 2-edge connected, and 22  H?"6@ NNN?Ndf  s *A L'??]Px e L'`  c $A( O'??MU O'!  Tk ?"6@ NNN?N17  _-If G has girth at least 4, then we are done !..H  0޽h ? fff̙___PPT10i.fi2+D=' = @B + TP ##0,..J#(  2  N?"6@ NNN?NP 2  N?"6@ NNN?N@w2  N?"6@ NNN?N92  N?"6@ NNN?Nw2  N?"6@ NNN?N" z |2  N?"6@ NNN?NPFB @ ND?"0@NNN?N fB   ND?"0@NNN?N} B   ND?"0@NNN?NwPB   ND?"0@NNN?NJJ@B   ND?"0@NNN?N}G}B   ND?"0@NNN?Nf6 f  <?"6@ NNN?N 2  N?"6@ NNN?N/ 2  N?"6@ NNN?N}2  N?"6@ NNN?Nc2  N?"6@ NNN?NZ8B  ND?"0@NNN?NB  ND?"0@NNN?N0B  ND?"0@NNN?N-  Tv ?"6@ NNN?N k9If not H1, H2, then find a (11, 3)-coloring by induction.::  Td{ ?"6@ NNN?N ) 3a  T< ?"6@ NNN?N]} 3b  T ?"6@ NNN?N]G} 3cr  Bu ?"6@ NNN?N Z F 9z f   z 9 f 2  N?"6@ NNN?N Yf 2  N?"6@ NNN?Nj V 2  N?"6@ NNN?N . O 2  N?"6@ NNN?Njz  2   N?"6@ NNN?Nm 8 2 ! N?"6@ NNN?N9 f B "B ND?"0@NNN?NY@ j" B # ND?"0@NNN?NY9 P B $ ND?"0@NNN?N@ B % ND?"0@NNN?N  B & ND?"0@NNN?N9 9 B ' ND?"0@NNN?N " " lF  j  (  j  ) TT ?"6@ NNN?Nt [  3a * Tt ?"6@ NNN?N~ s j  3b + T  ?"6@ NNN?N   3c , Tܖ ?"6@ NNN?N  3c - T ?"6@ NNN?N   3a . Tܜ ?"6@ NNN?N{ g"  3bH  0޽h ? fff̙//___PPT10/.aic+D.' = @B D.' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D ' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(++0+ ++0+ ++0+ ++0+ +XP )) ,<4)(  2  N?"6@ NNN?NP 2  N?"6@ NNN?N92  N?"6@ NNN?N" z |2  N?"6@ NNN?NPFB   ND?"0@NNN?N} B   ND?"0@NNN?N}G}B   ND?"0@NNN?Nf6 f  <?"6@ NNN?N ,$@  02  N?"6@ NNN?N"|",$@ 02  N?"6@ NNN?NZ8,$@ 0B  ND?"0@NNN?N,$@ 0B  ND?"0@NNN?N9,$D 0`  TL ?"6@ NNN?N ,$  0 j8If not H1,H2, then find a (11, 3)-coloring by induction.99)  T ?"6@ NNN?NV6v,$  0 3cr  Bu ?"6@ NNN?N ,$D   0 / S T0e0e    BCDDEF A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||DD"j "D@  s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab9bl *  8 * ,$D  02 0 N?"6@ NNN?N) J 2 1 N?"6@ NNN?N  l 2 2 N?"6@ NNN?N^ U 2 3 N?"6@ NNN?N*)  B 4 ND?"0@NNN?NJV m B 5 ND?"0@NNN?NV V B 6 ND?"0@NNN?N ? t ?  7 c Z0e0e    BCDDEF @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||DD"j "D@  s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab4  ) 9 TH ?"6@ NNN?N  ,$   0 3c) : T  ?"6@ NNN?Nz W / ,$   0 3c+ ; T ?"6@ NNN?N  ,$   0 5c+3F < T4 ?"6@ NNN?N a2 ,$  0 P c+6H  0޽h ? fff̙..___PPT10m..aic+OhD,' = @B D,' = @BA?%,( < +O%,( < +D ' =%(D[ ' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8DN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*9D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*9D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*:D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*:D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*<+P+0+ ++0+ ++0+9 ++0+: ++0+; ++0+< +{P @,{(  C  T ?"6@ NNN?N<B OIf the reduced graph is H1 or H2, then the We can find a (7,2)-edge coloring. PPH  0޽h ? fff̙80___PPT10.ji@P `,(  X  C 0A,Hgraphs (3)CDS,H  0޽h ? fff̙___PPT10i.ji41+D=' = @B +P ,(  V  C .A-h1convertsq% H  0޽h ? fff̙___PPT10i.ji+D=' = @B +x P , x(  r  S -XA`}   x  c $.X p   0 6A) '? ?.X: '   N?"6@ NNN?N]G   N?"6@ NNN?NW  T ?"6@ NNN?NPup  <    N?"6@ NNN?N% @6   0 6A* '? ?.X~p #  ' H  0޽h ? fff̙80___PPT10.ti!P , (    T  ?"6@ NNN?N4 HTheorem [Thomas Kraiser, Daniel Kral, Riste Skrekovski]IIP    N?"6@ NNN?N}nO Z(If G is cubic and has large girth, then ))f  s *A. '?? aq  '2  T ?"6@ NNN?N W ,$ 0 <   c $A+ '??f 7 8 '$@ 0X   TT ?"6@ NNN?N  2,$ 0 b depends on the girth.H  0޽h ? fff̙W O ___PPT10/ .ui $h+uc.D' = @B DN' = @BA?%,( < +O%,( < +D' =%(D-' =%(Db' =K@BB BBPB0B%(/%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* +p+0+ ++0+  +'P xp,(    T" ?"6@ NNN?N TTheorem [Thomas Kraiser, Daniel Kral, Riste Skrekovski, Xuding Zhu]UUb    N0*?"6@ NNN?N}n R For any graph G of large girth, !!f  s *A/ '?? M  '  T/ ?"6@ NNN?N W  < `  c $A+ '??f 7  '$  T2 ?"6@ NNN?N  2 b depends on the girth.H  0޽h ? fff̙W O ___PPT10/ .ui $h+D' = @B DN' = @BA?%,( < +O%,( < +D' =%(D-' =%(Db' =K@BB BBPB0B%(/%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*+p+0+ ++0+ +P rj, (  R  T> ?"6@ NNN?NuO ^Conjecture 1: For any graph G of maximum of maximum degree k, either__`  c $A0 '??f v '  T@ ?"6@ NNN?N7%  O or`  c $A; '?? &  ';  TG ?"6@ NNN?N" kB,$ 0 ETrue for k=1, 2, 3.D   TLL ?"6@ NNN?N ,$ 0 NIf it is true, is it sharp ?H  0޽h ? fff̙  ___PPT10 .vi8+4InD" ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* +p+0+ ++0+  + P -,(    TW ?"6@ NNN?NA Z(For any k, find a k-regular graph G with))`  c $A< '??3c  'h  TD\ ?"6@ NNN?N * ,$ 0 r@For k=2, the 5-cycle. For k=3, the Petersen graph. For k =4, ???AAH  0޽h ? fff̙___PPT10x.vi@} +T^D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*+8+0+ +ZP 0- Z(    T0f ?"6@ NNN?N  PIs there any relation between `  c $A? '??  '  Tk ?"6@ NNN?N  6 and+  To ?"6@ NNN?N'  i for cubic graphs other than `  c $A' '?? |  '`   c $A= '??    '5   Tt ?"6@ NNN?N   sAIn particular, is it true that cubic graphs of large girth haveBB`   c $A@ '??  'H  0޽h ? fff̙80___PPT10.wi7y?P P-(  '  Tu ?"6@ NNN?N#  eThe flower snarks N  C &AAj3j5j7   TH ?"6@ NNN?N0u< 4J3  Th ?"6@ NNN?N2G R 4J5  TT ?"6@ NNN?N9PY 4J7H  0޽h ? fff̙80___PPT10.wi@1P 80p- (    TH ?"6@ NNN?NW  CTheorem [Steffen]`  c $AB '??  'E  T ?"6@ NNN?Nt ,$ 0 O(I conjecture equality holds)l u 9v   u9v,$D 0  T ?"6@ NNN?Nu t  MQuestion: Is it true that h  c $AC '??f C v 'B   TD?"0@NNN?N 9    T ?"6@ NNN?N @ 9` 53 ?H  0޽h ? fff̙  ___PPT10y .xi`+vŹD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* +8+0+ +P --(  +  TX ?"6@ NNN?N: i7We see GAPS among circular chromatic indexes of graphs.88n  T ?"6@ NNN?N:,$ 0 xF(5/2, 3), (7/3, 5/2), (9/4, 7/3), & $$N  T ?"6@ NNN?N,$ 0 X (11/3, 4),     T ?"6@ NNN?N T ,$  0 Conjecture: Among the rational numbers which are the circular chromatic indexes of graphs, there is no strictly increasing bounded infinite sequence. H  0޽h ? fff̙___PPT10y.xi+PoQ?D' = @B D`' = @BA?%,( < +O%,( < +D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ +GP 00-ADc0(  B  ND?"0@NNN?NkC  Tq ?"6@ NNN?NQ? > Real numbers  l    2B,$D 0B  ND?"0@NNN?N  B  ND?"0@NNN?NV V B   ND?"0@NNN?N } B   ND?"0@NNN?N9 f B   ND?"0@NNN?N]9 ]f B   ND?"0@NNN?NFO Ff B   ND?"0@NNN?N  B  ND?"0@NNN?Nz z} B  ND