ࡱ> nV2LEbqLpPNG  IHDR 0Gr]gIFxNETSCAPE2.0$NPPLTEO~`ܬ׹B(+]!?+T@^Db6 ! ȸhhhPPP888 иȨАppPP0p@ 8 ݚևtW:]:y`@h؀XP8@(0谸x8P(pXH8(xph`XP((`0ИpȘHx `PpH`@P0H(8 0 ὧӨœo~Si7T@0` @ Ͽ?_ϟ߿@Wo?_Ͽ?_@Wo?_ϟ߿f+vذF*IZP’]< RYXn $c2IC,{E%(EM)9[?.<' ~>m;mySxn "|C13 <!偉ω{!S76M9w)FDS("Q&x<^uvQ+1r"^y@@kJ-OAQQU0d$=䜉+-;,uL2C jPj2dZ+c.Ʋ\Cq',l){AxVo 㛖WyGPbֺZ=T0 8⩿(z@Y;4@"݇tjC)PJ 躥. msOGMSOFFICE9.0GIF89a 03f3333f333ff3fffff3f3f̙3f3333f3333333333f3333333f3f33ff3f3f3f3333f3333333f3̙333333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffffff3fffffff3fff̙ffff3fffff3f̙3333f33̙3ff3ffff̙f3f̙3f̙̙3f̙3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3f! NETSCAPE2.0! , 0O~`ܬ׹B(+]!?+T@^Db6 ! ȸhhhPPP888 иȨАppPP0p@ 8 ݚևtW:]:y`@h؀XP8@(0谸x8P(pXH8(xph`XP((`0ИpȘHx `PpH`@P0H(8 0 ὧӨœo~Si7T@0` @ Ͽ?_ϟ߿@Wo?_Ͽ?_@Wo?_ϟ߿f+v0!IYR3V-U´Pv #q*Jv%ˠ,]Wլ:eɰ%ݪSڵ@x!F )VHr )P.L"%L]tf"@JT(;@Ұ3)L4;) (Q"ƃFPqؾLD|v'qNSy_ԩ PQ|%W^& 8m|`l&x'uIZx."$zbD,b $ȡ:62.(c'^B5XPH$?.&M#\F\i&c 70kE9Y;YE@ǧ@ ڧ0(24(dxQhDh *jJdQZ@*n jA֊)k*lAJkA! , 0O~`ܬ׹B(+]!?+T@^Db6 ! ȸhhhPPP888 иȨАppPP0p@ 8 ݚևtW:]:y`@h؀XP8@(0谸x8P(pXH8(xph`XP((`0ИpȘHx `PpH`@P0H(8 0 ὧӨœo~Si7T@0` @ Ͽ?_ϟ߿@Wo?_Ͽ?_@Wo?_ϟ߿f+v%N*ӵ;'5rG Ua];inGl4b ]Z8V.Mcͦ{8a*|P"֩WH0) 0D@=;DSH@%B88m %W{@߃(^Dmw7*:GK @!? trz& ɂCx|6x lB&|xr!n("P'޷⊗xҢF"hb# .6 '56^R$C0$"~BGbMNy%0衍V~9llnp 9u Fu 's֩@6P9 h0rFjj r@)Xjp)R2Z:xQCdjj+Jkj ۨ8KZ;IENDB`FzFE-Qڛb>"JFIFKKMSO Palette C   ")$+*($''-2@7-0=0''8L9=CEHIH+6OUNFT@GHEC !!E.'.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE"!!1AQqa!1 ?F3]'IƓe5,J$3?O 5LXyYL4tO%fxM\DXL`: a2HS=rtd+f\UtV5m7L?cf7 `Zx5\kn@b2;NPNG  IHDR99sgIFxNETSCAPE2.0$NPPLTEնjzzII׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\(؃Q DEH>,sR̈E%pB4̉D,[8$ <~\ɕ4MJ."$S^:V뚳v5!"֧&:yoi'z| :fL:QXb@bݛ[H珑蘄2˭inm{ ٶCc"i\sќ7 L E\8&xNQ3\ܤ!M>y=?q3a;JD2QMϡ ̇'Xlُkξî(ҢةhP.HS"4}{[hFL"%JZGu>I(@DM OSؖ}qjnBZ[Ihv/w+sVBeka<0\()%W}=d| =gh]b0al+p0(pxB m /B(['x6vP9.Dp89" puB $@yBp(msL iep@W&Alw꘥ghýK掲ਣ (D]z({N o=觠jp ڪn9`@8+7AڊAlu@,Z@R@b \^ mrq؂kzAn. 8`G-ĵ{@/J ˰Ll W1+AL!/'+)'3@5|@pȁ"p!;SP,"Mt='pt& p@Xgmu[! *,99նjzzII׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\()%W}=d| =gh]b0al+p0(pxB m /B(['x6vP9.Dp89" puB $@yBp(msL iep@W&Alw꘥gsK(t) Q9z@pZJj]Yghf[9jxN0+`y+DAQz;-]{Q+YfY.ekA+qB <̰=qi( i (2,3Ӭ8r pHJ1 ( 8mp\g! ),99նjzz׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\()%W}=d| =gh]b0al*p0'ph m /B([&h6r@9.Dp8b9 pqBФ #@iBp(msL iep0W&Alw꘥g(* pB%{ZFjoVj)xۦ9j@tҮJ*\:y`+@Ahj>{pZJzYrm`+.y@pΊKܹk.ƻozoz@AzA I,[LBgB$r |3A+ES P#e@! #,99նjzz׻w~]ּcUՃ˥;zʭP1HZxT!e>Ihv/w+sVBek\()%W}=d|v %` 0i aoL !*І$l3Vp.&p$#c9#A;XdG"١gA)86 pZB"y of{\ۚ &.@lA~F'T :y`(<.iZ/z&gX2)pˁp:q  \.;lr,r@#X ,4{n ]{ ] 0/٫+ ,dKL 10qŏp1$ $,.! *,99նjzzII׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\()%W}=d| =gh]b0al+p0(pxB m /B(['x6vP9.Dp89" puB $@yBp(msL iep@W&Alw꘥gsK(t) Q9z@pZJj]Yghf[9jxN0+`y+DAQz;-]lۭ 㒻gdf|Ѐp֪Kܻ뮼  o~+A~A i{̱B&k0B( rB-\3t9D R3*BFsI/}lMs$X'e@! ),99նjzz׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\()%W}=d| =gh]b0al*p0'ph m /B([&h6r@9.Dp8b9 pqBФ #@iBp(msL iep0W&Alw꘥g(* pB%{ZFjoVj)xۦ9j@tҮJ*\:y`+@Ahj>{pZJzYrm`+.y@pΊKܹk.ƻozoz@AzA I,[LBgB$r |3A+ES P#e@! #,99նjzz׻w~]ּcUՃ˥;zʭP1HZxT!e>Ihv/w+sVBek\()%W}=d|v %` 0i aoL !*І$l3Vp.&p$#c9#A;XdG"١gA)86 pZB"y of{\ۚ &.@lA~F'T :y`(<.iZ/z&gX2)pˁp:q  \.;lr,r@#X ,4{n ]{ ] 0/٫+ ,dKL 10qŏp1$ $,.! ),99նjzz׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\()%W}=d| =gh]b0al*p0'ph m /B([&h6r@9.Dp8b9 pqBФ #@iBp(msL iep0W&Alw꘥g(* pB%{ZFjoVj)xۦ9j@tҮJ*\:y`+@Ahj>{pZJzYrm`+.y@pΊKܹk.ƻozoz@AzA I,[LBgB$r |3A+ES P#e@! *,99նyjzz׻w~]ּcUՃ˥;zʭPl<1HZxT!e>Ihv/w+sVBeka<0\()%W}=d| =gh]b0al+p0(pxB m /B(['x6vP9.Dp89" puB $@yBp(msL iep@W&Alw꘥g(* pB%{ZFjoVj)xۦ9j@xѮN*`:y`+DQhj,CJz&%\6ikdk|Ѐ&nB˯[@+A T[<@k1A( <@%ܲ+*P!"Lt ۜB&! D- l! &,99նjzzIIw~]ּcU˥;zʭPl<1HZxT!e>Ihv/w+sVBa<0\({o 0tAD  t#tsܱ !< dbGm )0spD-=! %,99նjzzIIw~]ּcU˥;zʭPl<1HZxT!e>hv/w+sVBa<0\(;oK0tA4 t#4cq wq dbGPmT%Pm0@=! &,99նjzzIIw~]ּcU˥;zʭPl<1HZxT!e>Ihv/w+sVBa<0\({o 0tAD  t#tsܱ !< dbGm )0spD-=! %,99նjzzIIw~]ּcU˥;zʭPl<1HZxT!e>hv/w+sVBa<0\(PҺ ]|] 09,S\k\B f@[О.@˲, p<|@! &,99նjzzIIw~]ּcU˥;zʭPl<1HZxT!e>Ihv/w+sVBa<0\({o 0tAD  t#tsܱ !< dbGm )0spD-=! ,99նjzzIIw~]ּcU˥;zʭP1HZxT!e>hv/w+sVB\(Ihv/w+sVBeka<0\()%W}=d| =gh]b0al+p0(pxB m /B(['x6vP9.Dp89" puB $@yBp(msL iep@W&Alw꘥g(* pB%{ZFjoVj)xۦ9j@xѮN*`:y`+DQhj,CJz&%\6ikdk|Ѐ&nB˯[@+A T[<@k1A( <@%ܲ+*P!"Lt ۜB&! D- l;OIENDB``!YE} E bkSV`R@ PV(+'xcdd``dd``baV d,FYzP1n:,B@?b ZUXRYplo?`'0LY ZZǰ@px9@]j mUg5*g$|F0;V Ps%\Ps073LH^&I9 @3\OuJ.hscа``ⴃI)$5$ŠQ _ 1#s3ȄJtw dnpenR~CP@K!2dblR^ #4"kTb wl ~ s+N `CXJ;LLJ% 1.E.b΃G L(`!ZDe]y0*6}Tsxcdd`` @c112BYL%bpuPhqS}ĪfO8+r;qQLY@& M5,Blxef jKUrvHbLDNZ@)qĜXTR `y;yqj6"iH3% ybH~n5A V&Lj́e%{k3@eG$j@$bh!7n6X.pIQN e2oi!TI*8.Q“nQ`xԱgcF~\vn{ÿʰ$ 4 #x=^ZK#6z(:sA):m"$-TFp=$dQP*=mTʁDOh֩*CH2Oz԰d$vq[V0@ @$`HӱlA ;M@: t7XR H2?\iI!@*ON s< M%&A$yHSxp:DdT1747oD3$`QQF]8Ted0b@;mz‚P`zWkVW6)bdH *rwڃK$O'>>@`&-EC` pW.g6XevlU1DbPŭG΂0ehYPC8GnߚW6*̤?ږNaJC`P(F,aAcfVwFa $g('M]m|AH6vjIXC 8SuiHA'f}~SQRmNwrZg41?B 3;} `1t{iVEkn N6iъm7RN 2$ L}h:ڋb#h+4.l-Q7+^*We@ V-$dS0~,mf@Hi ` W(fP#opӤ9#z!S }l+tLAC0;n}Б$5A,w ~tT;ϯ'7$ #4 *)sYeaH䁸,VȀp*ϼH8&?@BQh0K@N70v#<ʺ|mMn '8%T 5Hm9KmRS|:$vYDV 21el gT[b+Yy rB` GTR rPñ{2J iQV@6<|]oba aOU2` L[V Ib /CR `!x ~eӘD?@h<`(+Fxcdd``Ned``baV d,FYzP1n:, B@?b Xzjx|K2B* R. KXB2sSRs=R~/g`0`Ƞ T+@Z< ׄY@| l3Ma(w r9bL@rGN1e$_s斱Lp/y梻s3Ȅtl[9ĺ׏dn24ֹ)P=Ĥ\Y\ bPd+_ 1O[T T#@Z ?ׄ_~&I9 \T ᰟD@rG [ sI\F L ܠ*'\t>`ͽ(+w8`׍dn24ֹ)P=Ĥ\Y\˰d.P"CXpx!Āf~7`!{ vZ%` 0Ixcdd``Ned``baV d,FYzP1n:lB@?b 10sC0&dT20 KXB2sSRsnX>?HT T@Z*a6V c5W`iTb dpenR~,Q./p? #df sI\FL ܡ*'\t>`ͽ(P&U;027 \^h 0y{@iI)$5a\E.!E`C`!K{j:f-˜@2Hxcdd`` @c112BYL%bpu{piUsi#.F5 +p]6b;F&&\y @ ]` _ v0@f~1)8`!6BM>bޒ XJxcdd``> @c112BYL%bpu @c112BYL%bpu"^b$b2;N@2$E} E bkSV`aZ2$[ N^-K{ \2$ZDe]y0]R$qܣ!h=;h1_2$ ~eӘD?@h<m2$co2$ vZ%p2${j:f-˜Sr2$koB=#Zjps2$BM>b>t2$%" W,z{ -u# lAA5%"  f3@ 3܇ʚ;ʚ;g4;d;dȅ 0ppp@ <4dddd k 0Tx 80___PPT10 pp?  %/$Game chromatic number of graphs B$3T|O|O -Nxbxex[@b c                     "  #  02-tree 1  48!:"W7<$  >&?'@(A)D+E,F-H.I/J0K1L2M3P4R5T6*/   P%&   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> h``X(  X X 6 " `}  L cN NN}/kGrjL#j_   X 0L " `  |0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\   X 0(& "^ `  X* X 0 & "^  & Z* X 0& "^ ` & Z*p X JA޽h ?v}"` 3380___PPT10.IP@  -!|1X-  0L0 P<(  ~  s *&X@ & ~  s *&Xc t4 & p  JA޽h ?e^}"` 33___PPT10i.P.+D='  = @B +   0 @*}P*(  2  H%?"0@NNN?N3  Two players: Alice and BobL 2 CG3CG̙ l   iw ,$D 0n2 ; 0"`Pn2 < 0"`-  n2 = 0"` ` n2 > 0"` c n2 @ 0"`w-ln2 A 0"` Q n2 B 0"`h2 H s *"`0  ON I S xH# N J S   c N L S  CN M S N N S }; {N O S = N R S -0 N S S wlN T S -] h2 U s *"`  h2 V s *"`~ @4 h2 W s *"`  TB [ c $D S  N _ S  CN ` S   N a S r c N b S 6 rN d S  ~ N e S  3 N f S   N g S  & N h S  4K { j <"P  ,$ 0 GA set of colors0Rl  J | J,$D 0tb l 6"`` nb r 0"` nb s 0"` nb t 0"` JP w S (A pic2P z S (A p75s@t &  { <)P- ,$ 0 A A graph G 0 H  0޽h ?/@B<I<>JB=L B;M ;=N <;O@HR@STVW_>V`<AaAWbHVd =Ae"UAf$;Wg&=Hh 3f  ___PPT10 .E+:+D'  = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*{%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*i%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*j%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(+p+0+j0 ++0+{0 +yD   0  -zH(  H F   KH w h2 LH s *"`Ph2 MH s *"`-  h2 NH s *"` ` h2 OH s *"` c h2 PH s *"`w-lh2 QH s *"` Q h2 RH s *"`h2 SH s *"`0  ON TH S xH# N UH S   c N VH S  CN WH S N XH S }; {N YH S = N ZH S -0 N [H S wlN \H S -] h2 ]H s *"`  h2 ^H s *"`~ @4 h2 _H s *"`  TB `H c $D S  N aH S  CN bH S   N cH S r c N dH S 6 rN eH S  ~ N fH S  3 N gH S   N hH S  & N iH S  4K {lb lH 6"` lb mH 6"` lb nH 6"` lb oH 6"` JJ pH C "A pic2J qH C "A p75s@t & 2 sH 0"` A O,$D  02 tH 0"` m # c ,$@  02 uH 0"`]  ,$D   02 vH 0"`@ ,$D   0 wH 0"`P },$D 0 xH <CP ,$ 0 n6Adjacent vertices cannot be colored by the same color.7072 yH 0"`@ ,$D  0H H 0޽h ?/@%RHMHTH&MHOHUH'RHNHVH(RHLHWH)LHNHXH*MHLHYH+PHSHZH,PH[H-\H.^H_HaH/OH^HbH0MHQHcH1QH_HdH2SH^HeH3NHQHfH4]HQHgH5LH_HhH6NHSHiH Z\l[]kfEoI.A.___PPT10!..Iq+TD-' FP= @B Dx-' = @BA?%,( < +O%,( < +D' =%(DR' =%(D' =4@BBBB%()?)?DN' =.7 BBBBBOM 0.00659 -0.00624 L 0.31372 -0.07976 *3>*B ppt_xB ppt_y=0BBAA<*pHD' =%(D' =%(D' =4@BBBB%()?)?DL' =.7 BBBBBMM 0.31372 -0.07977 L 0.0224 -0.01688 *3>*B ppt_xB ppt_y=0BBAA<*pHDg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*sH%(D' =-s6Bwipe(down)*<3<*sHD' =%(D^' =%(D' =4@BBBB%()?)?DZ' =."7 BBBBB[M 1.66667E-6 -4.62428E-6 L -0.29132 0.13642 *3>*B ppt_xB ppt_y=0BBAA<*qHD[' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*tH%(D' =-s6Bwipe(down)*<3<*tHD<' =4@BBBB%()?)?D' =.X7 BBBBBM -0.29132 0.13641 C -0.17101 0.08231 -0.05052 0.0282 -0.00243 0.00739 *3>*B ppt_xB ppt_y=0BBaA<*qHD' =%(D\' =%(D' =4@BBBB%()?)?DX' =. 7 BBBBBYM 3.88889E-6 5.83815E-6 L 0.29132 -0.14681 *3>*B ppt_xB ppt_y=0BBAA<*pHD' =%(D' =%(D' =4@BBBB%()?)?DN' =.7 BBBBBOM 0.29132 -0.14682 L 0.00781 -0.01063 *3>*B ppt_xB ppt_y=0BBAA<*pHDg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*uH%(D' =-s6Bwipe(down)*<3<*uHD' =%(D`' =%(D' =4@BBBB%()?)?D\' =.$7 BBBBB]M 4.16667E-6 -4.16185E-6 L -0.20469 -0.08393 *3>*B ppt_xB ppt_y=0BBAA<*qHDa' =%(D ' =%(DB' =4@BBBB%()?)?D' =.^7 BBBBBM -0.20468 -0.08393 C -0.11683 -0.04555 -0.02881 -0.00693 0.00643 0.00925 *3>*B ppt_xB ppt_y=0BBaA<*qHDg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*vH%(D' =-s6Bwipe(down)*<3<*vHD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*wH%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*xH%(D' =%(Dd' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*wH%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*vH%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*yH%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*yHD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*yH+8+0+xH0 +1   0 %@3:Pm(  P F   P w h2 P s *"`Ph2 P s *"`-  h2 P s *"` ` h2  P s *"` c h2  P s *"`w-lh2  P s *"` Q h2  P s *"`h2  P s *"`0  ON P S xH# N P S   c N P S  CN P S N P S }; {N P S = N P S -0 N P S wlN P S -] h2 P s *"`  h2 P s *"`~ @4 h2 P s *"`  TB P c $D S  N P S  CN P S   N P S r c N P S 6 rN P S  ~ N  P S  3 N !P S   N "P S  & N #P S  4K {&F  J %P  Jnb &P 0"` nb 'P 0"` nb (P 0"` nb )P 0"` JJ *P C "A pic2J +P C "A p75s@t & f2 .P 0"`]  f2 /P 0"` A O2 0P 0"` ,$D 0f2 1P 0"` m # c f2 2P 0"`@ 2 3P 0"`P=,$D 02 4P 0"`l,$D 02 6P 0"``z 0 ,$D 0f2 7P 0"`w-2 8P 0"`C ,$D 0 9P <cP ( ,$ 0 C Game over ! 0 2 :P 0"` - ,$D 0H P 0޽h ?/@ PPPP PP PPP PPPPPPPPP P PP PP P PPP PPP P PP PPP PPPP P PP P!PPP"PP P#P Z\l[]kfEo___PPT10.Iy+sD'  = @B DM' = @BA?%,( < +O%,( < +Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*3P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6P%(Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*0P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8P%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9P%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*:P%(D' =-6B'blinds(horizontal)*<3<*:PD' =%(D' =%(D0' =4@BBBB%(D' =0l9 CCBB*<3<*7PD' =%(D0' =4@BBBB%(D' =0l9 CCBB*<3<*3PD' =%(D0' =4@BBBB%(D' =0l9 CCBB*<3<*0PD' =%( D0' =4@BBBB%(D' =0l9 CCBB*<3<*1P+8+0+9P0 +#   0 SK0 +L(  L #L <uP:r a)Game is over when one cannot make a move.*0*  $L <XzP F,$ 0 k#Either all the vertices are colored,$0I %L <ԀPC ,$ 0 aOr there are uncolored vertices, but there is no legal color for any of the uncolored vertices,b0_z g' &L g7,$D 0` 'L 0' (L <ЅPtg F Alice wins 0 z  J  )L  J ,$D 0` *L 0 Y J  +L <P   DBob wins 0 H L 0޽h ? Z\l[]kfEoxp___PPT10P.I+B#style.visibility<*$L%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%L%(D' =%(DF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*&L%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&LDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)L%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*)LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*)L+p+0+$L0 ++0+%L0 +   0 tz(  t  t <P XAlice s goal: have all the vertices colored.,-0. t <P_  zBob s goal: to have an uncolored vertex with no legal color.,>0 1H t 0޽h ? Z\l[]kfEo___PPT10i.I+B#style.visibility<*`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!`%(+p+0+`0 ++0+!`0 +-    0 !dj(  d& d <P1t`_ BIt depends on the graph G, and depends on the number of colors !\C0 d <ػPati ,$ 0 6Given a graph G, the game chromatic number of G is the least number of colors for which Alice has a winning strategy.06  d 0A ??7 _ 8 $@ 0 !d s * p,$D 0H d 0޽h ? Z\l[]kfEo___PPT10.I+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!d%(+8+0+d0 +    0 $l(  l l <Z& RTo prove that (0 ` l c $A ??@    l <P1t_ ? one needs to prove the correctness of a sentence of the form:(@0??  l s *A ??W8 $@ 0 !l <P * ,$ 0 `(MA: a move for Alice; MB: a move for Bob)0) "l <xP6 *j ,$ 0 a%A hint that the problem is difficult. &0&H l 0޽h ? Z\l[]kfEo___PPT10.I+B#style.visibility<* l%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!l%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"l%(+p+0+!l0 ++0+"l0 +   0 0*GG(  l   L,$@ 0.  <$P <Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]=0=>  h  c $A .??M %  .  <^&c B0   <P UFor any forest F,(0P  0B S  K N h2 ? s *"`S h2 A s *"`V@ w%   Fw%  n2  0"`Z  T B c $ u T  c $ < n2   0"`z 0 e n2   0"`w -e n2   0"`  T  c $ n2  0"`% z T B c $ 3 n2  0"`/ s T  c $_ J n2  0"`? e T  c $ X Z h2 7 s *"` a E h2 8 s *"`3 f N : S z N ; S  E h2 @ s *"`S f N B S F * n N C S f N D S K H  0޽h ?   7:78;89<85>7@B@?C!@AD Z\l[]kfEo"___PPT10.I+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G%(+   0 @/@(    < 0C e-Theorem [Guan and Zhu, J. Graph Theory, 1999].0.  B o"`  C  'F N      Bl   ;G0  6   B7.$G  6  A)$G  6   W $G    6d# @   A($G   6) & A$G   6d!  d  Cg&G   6H2 N   Cc&G  <6 s  For any outerplanar graph G,(0  `2  s *"`BF @ S '  F  S BJ `2  s *"`f `2  s *"` G5 `2  s *"` : `2  s *"`9 - `2  s *"` X F  S '; F  S   `2  s *"` ^ BF @ S  `2  s *"`9 @ F  S B 9 `2  s *"` ` 8 F  S  { F  S h =U& F  S f - F  @ S  @ F ! S 5  `2 2 s *"` ~ `2 3 s *"`CR `2 4 s *"` F F 5 S ^ qF 7 S p~ ^F 8 S c  F 9@ S F & l [ f  @[ f ,$D 0h2 < s *"` @ [ f  ?[ f ZB : s *D   T ;B c $ M B T = c $ M { f T > c $  u [ T H  0޽h ?/@    !523724849<;!<=#<> Z\l[]kfEo___PPT10.I+B#style.visibility<*@%(D' =%(D' =%(D5' =4@BBBB%(D' =-s6Bwipe(down)*<3<*@D' =1:Bhidden*o3>+B#style.visibility<*@%(+@   0 *} /(    0H M7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N   @   BHN   ;G0  6xP  E3k+2.$G  6U  A)$G  6TZ   W $G    6(` @   A($G   6xd & A$G   68i  d  Cg&G   6m N   Cc&G  0r P  For any partial k-tree G,l0CCC T H u  XpTW6 <$ 0   z   y \ ,$D 0t2 V 6"` D2 W B?"`5ZB X s *DCp6pl t  z]  ,$D  0t2 Z 6"`_ T ZB [ s *Dc ZB \ s *D&tc z  2%  ] = d ,$D  0n2 ^ 0"`pS % TB _ c $D S p TB ` c $D 2Z l  V  {  ,$D  0t2 b 6"` GV ZB c s *D  ZB d s *DF  z G}j  e  ,$D  0TB fB c $D}c 2 g < "`G6   L    TB h c $D j z  @  }  @ ,$D   0ZB jB s *DZ rZB k s *Ds @ rt2 l 6"`Ez =  | = ,$D   0ZB n s *D= } a ZB o s *D] } a t2 p 6"`"  b x B臊 ,$  0 *A partial k-tree is a subgraph of a k-treeT+ CCCCC H  0޽h ? Z\l[]kfEo((___PPT10`(.I+B#style.visibility<*T%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*y%(D' =%(DF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*z%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*zD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*zDn' =%(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' =%(DF' =%(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<*{Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*eD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*eDd' =%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*eD' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*eD' =1:Bhidden*o3>+B#style.visibility<*e%(D' =%(DF' =%(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' =%(DF' =%(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<*|DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(+p+0+T0 ++0+x0 +   0 )!,8q(    < C ? Theorem [Zhu]  B o"`  C  (F N    5  Bl   ;G0  6D  C17.$G  6  A)$G  6|   W $G    6 @   A($G   6d & A$G   6$Ɗ  d  Cg&G   6ʊ N   Cc&G  <ϊ "YD [For any planar graph G,(0`2  s *"`BF @ S '  `2  s *"`f `2  s *"` G5 `2  s *"` : `2  s *"`9 - `2  s *"` X F  S   `2  s *"` ^ BF @ S  `2  s *"`9 @ F  S B 9 F  S   F  S h =U& F  S f - F  @ S  @ F ! S 5  `2 " s *"` 3 b `2 # s *"`p% `2 $ s *"`) T F & S   F ' S G D `2 0 s *"`  `2 1 s *"` =s F 2 S  c F 3 S BJJ F 4 S c = F 5@ S s 9 F 6 S X [ T F 7@ S ' F 8@ S b T H  0޽h ?/@$    ! "#&"$'0203014151617 "8 Z\l[]kfEo___PPT10i.I  f  s *A ??    B4   B0   B  UFor any forest F,(0  <t  JProof:(08 7@I ]@7In2   0"`M  s n2   0"`  ZB   s *Dw  T  c ${ n2  0"` h ZB B s *D; K n2  0"` n2  0"`4 T  c $4 T B c $ f  n2  0"`  n2  0"`F  T w%   # @ h2  s *"`Z  N B S  u N  S  < h2  s *"`z 0 e h2  s *"`w -e h2  s *"`  N  S  h2  s *"`% z N  B S  3 h2 ! s *"`/ s N " S _ J h2 # s *"`? e N $ S  X Z h2 % s *"` a E h2 & s *"`3 f N ' S z N ( S  E h2 ) s *"`S f N * S F * n T + c $T T T , c $f  n2 - 0"`9  n2 . 0"`:9  n2 / 0"`-9  n2 0 0"`z 9 0 n2 1 0"` 9 n2 2 0"` 9 a n2 3 0"`9 T n2 4 0"`9 t n2 5 0"`9  T 6B c $ T T 7 c $  9 T 8 c $M HT T 9B c $ . T T : c $n n 9 T ; c $  9 T < c $ 9 T =B c $ 9 T > c $ 99 n2 ? 0"`7,n2 @ 0"`,n2 A 0"`,Kn2 B 0"`Z,n2 C 0"` , n2 D 0"` ,m n2 E 0"`" , n2 F 0"` ,n2 G 0"`,cn2 H 0"`s,)n2 I 0"`,IT JB c $ 4,T KB c $W UGT L c $ ,T M c $ uGT NB c $z ,T O c $ GT P c $0 = GT QB c $C ,T R c $9 ,T SB c $ GT T c $y ,h2 [ s *"`g  N \ S '  2 _ 0"`@j ,$D 02 ` 0"`  ,$D 02 a 0"`P  _,$D 02 b 0"` F ,$D 02 c 0"`9 - ,$D 02 d 0"` g ,$D 02 e 0"` ,$D  0H  0޽h ??`  !"!#$%' %&( & & %)* )+),-6.7/8091:2;3<4= 5>"-?J$.K&.AL'.BM(0CN)0DO*0EP+3FQ-3GR/5HS15IT2)[\ 33&___PPT10.@a+^dD' = @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<*`%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D ' =%(D' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*cD' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*cD' =1:Bhidden*o3>+B#style.visibility<*c%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*dD' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*dD' =1:Bhidden*o3>+B#style.visibility<*d%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(+u\  0 22@ej,(  F 7@I  @7Ih2  s *"`M  s h2  s *"`  TB  c $Dw  N  S { h2   s *"` h TB  B c $D; K h2   s *"` h2   s *"`4 N   S 4 N B S  f  h2  s *"`  h2  s *"`F  T w%   # @ h2  s *"`Z  N B S  u N  S  < h2  s *"`z 0 e h2  s *"`w -e h2  s *"`  N  S  h2  s *"`% z N B S  3 h2  s *"`/ s N  S _ J h2  s *"`? e N  S  X Z h2  s *"` a E h2   s *"`3 f N ! S z N " S  E h2 # s *"`S f N $ S F * n N % S T T N & S f  h2 ' s *"`9  h2 ( s *"`:9  h2 ) s *"`-9  h2 * s *"`z 9 0 h2 + s *"` 9 h2 , s *"` 9 a h2 - s *"`9 T h2 . s *"`9 t h2 / s *"`9  N 0B S  T N 1 S   9 N 2 S M HT N 3B S  . T N 4 S n n 9 N 5 S   9 N 6 S  9 N 7B S  9 N 8 S  99 h2 9 s *"`7,h2 : s *"`,h2 ; s *"`,Kh2 < s *"`Z,h2 = s *"` , h2 > s *"` ,m h2 ? s *"`" , h2 @ s *"` ,h2 A s *"`,ch2 B s *"`s,)h2 C s *"`,IN DB S  4,N EB S W UGN F S  ,N G S  uGN HB S z ,N I S  GN J S 0 = GN KB S C ,N L S 9 ,N MB S  GN N S y ,h2 O s *"`g  N P S '  2 Q 0"`@j ,$D 0& R <<* ,L` <Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]=0=>  ` S c $A ??   T <4,   B0  U <46  UFor any forest F,(0 V <@;  JProof:(02 W 0"`, ,$D 02 X 0"`P  _,$D 02 Y 0"`9 a ,$D 0 Z c $| { ,,$D 02 [ <|"`@j ,$D 02 \ <|"`, ,$D 02 ] <|"`9 z 0 ,$D 0 ^ c $|  . J ,$D 02 _ <|"`  ,$D  0 ` c $|f ,$D  02 a <|"` ,$D  0 b c $|_ 3 ,$D  02 c <|"`P  _,$D  0 d c $|* k ,$D 02 e <|"`9 ` ,$D 0 f c $|  / ,$D 02 i 0"`  ,$D 0H  0޽h ? !  "   #$ #%#& '0 (1 )2 *3 +4 ,5 -6.7/8'9D(E(;F(<G*=H*>I*?J-@K -AL!/BM"/CN##OP%W*Z&] ^'_ `(aXb)[d*e_f 33))___PPT10j). @+ID>)'  = @B D(' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i%(+q\  0 ;939`uy2(  F 7@I  @7Ih2  s *"`M  s h2  s *"`  TB  c $Dw  N  S { h2  s *"` h TB B c $D; K h2   s *"` h2   s *"`4 N   S 4 N  B S  f  h2   s *"`  h2  s *"`F  T w%   # @ h2  s *"`Z  N B S  u N  S  < h2  s *"`z 0 e h2  s *"`w -e h2  s *"`  N  S  h2  s *"`% z N B S  3 h2  s *"`/ s N  S _ J h2  s *"`? e N  S  X Z h2  s *"` a E h2  s *"`3 f N  S z N   S  E h2 ! s *"`S f N " S F * n N # S T T N $ S f  h2 % s *"`9  h2 & s *"`:9  h2 ' s *"`-9  h2 ( s *"`z 9 0 h2 ) s *"` 9 h2 * s *"` 9 a h2 + s *"`9 T h2 , s *"`9 t h2 - s *"`9  N .B S  T N / S   9 N 0 S M HT N 1B S  . T N 2 S n n 9 N 3 S   9 N 4 S  9 N 5B S  9 N 6 S  99 h2 7 s *"`7,h2 8 s *"`,h2 9 s *"`,Kh2 : s *"`Z,h2 ; s *"` , h2 < s *"` ,m h2 = s *"`" , h2 > s *"` ,h2 ? s *"`,ch2 @ s *"`s,)h2 A s *"`,IN BB S  4,N CB S W UGN D S  ,N E S  uGN FB S z ,N G S  GN H S 0 = GN IB S C ,N J S 9 ,N KB S  GN L S y ,h2 M s *"`g  N N S '  f2 O 0"`@j & P <r ,L` <Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]=0=>  ` Q c $A ??   R <t   B0  S <x~  UFor any forest F,(0 T <  JProof:(0f2 U 0"`, f2 V 0"`P  _f2 W 0"`9 a L X c $| { ,r2 Y <|"`@j r2 Z <|"`, r2 [ <|"`9 z 0 L \ c $|  . J r2 ] <|"`  L ^ c $|f r2 _ <|"` L ` c $|_ 3 r2 a <|"`P  _L b c $|* k r2 c <|"`9 ` L d c $|  / f2 e 0"`  f2 f 0"`9  2 g <|"`9  ,$D 0 h c $| J ,$D 02 i <|"` h ,$D 0 j c $| :K ,$D 02 k <|"` ,$D 0 l c $|x,$D 02 m <|"`2,$D 0 n c $|C,$D  02 o 0"`< ,$D  02 p 0"`,Z,$D  02 q <|"`,Z,$D  0 r c $| u=,$D  02 s <|"`9 : ,$D 0 t c $| / ,$D 02 v 0"` [ ,$D 0H  0޽h ?       !" ! #!$%.&/'0 (1 )2 *3 +4,5-6%7B&C&9D&:E(;F(<G(=H+>I +?J!-@K"-AL#!MN$U(X%[ \&]^'_V`(Yb)c]d*gh+j,kl-mYn.q&r/sit 33""___PPT10". @+gD"'  = @B DE"' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*j%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*n%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*o%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*q%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*r%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*s%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*v%(+NE  0 l:d:@|}<3(  <F 7@I < @7Ih2 < s *"`M  s h2 < s *"`  TB < c $Dw  N < S { h2 < s *"` h TB <B c $D; K h2  < s *"` h2  < s *"`4 N  < S 4 N  <B S  f  h2  < s *"`  h2 < s *"`F  T w%   <# @ h2 < s *"`Z  N <B S  u N < S  < h2 < s *"`z 0 e h2 < s *"`w -e h2 < s *"`  N < S  h2 < s *"`% z N <B S  3 h2 < s *"`/ s N < S _ J h2 < s *"`? e N < S  X Z h2 < s *"` a E h2 < s *"`3 f N < S z N  < S  E h2 !< s *"`S f N "< S F * n N #< S T T N $< S f  h2 %< s *"`9  h2 &< s *"`:9  h2 '< s *"`-9  h2 (< s *"`z 9 0 h2 )< s *"` 9 h2 *< s *"` 9 a h2 +< s *"`9 T h2 ,< s *"`9 t h2 -< s *"`9  N .<B S  T N /< S   9 N 0< S M HT N 1<B S  . T N 2< S n n 9 N 3< S   9 N 4< S  9 N 5<B S  9 N 6< S  99 h2 7< s *"`7,h2 8< s *"`,h2 9< s *"`,Kh2 :< s *"`Z,h2 ;< s *"` , h2 << s *"` ,m h2 =< s *"`" , h2 >< s *"` ,h2 ?< s *"`,ch2 @< s *"`s,)h2 A< s *"`,IN B<B S  4,N C<B S W UGN D< S  ,N E< S  uGN F<B S z ,N G< S  GN H< S 0 = GN I<B S C ,N J< S 9 ,N K<B S  GN L< S y ,h2 M< s *"`g  N N< S '  f2 O< 0"`@j & P< < ,L` <Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]=0=>  ` Q< c $A ??   R< <   B0  S< < ˍ  UFor any forest F,(0 T< <Ѝ  JProof:(0f2 U< 0"`, f2 V< 0"`P  _f2 W< 0"`9 a L X< c $| { ,r2 Y< <|"`@j r2 Z< <|"`, r2 [< <|"`9 z 0 L \< c $|  . J r2 ]< <|"`  L ^< c $|f r2 _< <|"` L `< c $|_ 3 r2 a< <|"`P  _L b< c $|* k r2 c< <|"`9 ` L d< c $|  / f2 e< 0"`  f2 f< 0"`9  r2 g< <|"`9  L h< c $| J r2 i< <|"` h L j< c $| :K r2 k< <|"` L l< c $|xr2 m< <|"`2L n< c $|Cf2 o< 0"`< f2 p< 0"`,Zr2 q< <|"`,ZL r< c $| u=r2 s< <|"`9 : L t< c $| / f2 u< 0"` [ z 9 v< 9,$D 0 w< <ߍ ]k aThere is a forest F such that(0h x< c $A ??@9 f2 y< 0"`,2 z< <|"`,,$D 0 |< c $| jU=,$D 02 }< 0"`C D ,$D 0H < 0޽h ?<<<<<<<<<<<<<<<<<<<<<<< << < < < < < < < <!<"< !< <#<!<<$<<%<.<<&</<<'<0< <(<1< <)<2< <*<3< <+<4<<,<5<<-<6<%<7<B<&<C<&<9<D<&<:<E<(<;<F<(<<<G<(<=<H<+<><I< +<?<J<!-<@<K<"-<A<L<#!<M<N<$U<(<X<%[< <\<&]<<^<'_<V<`<(Y<b<)c<]<d<*g<<h<+<j<,k<<l<-m<Y<n<.q<&<r</s<i<t<1z<s<|< 33  ___PPT10b . @+ҘD6 '  = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*z<%(Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*|<%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*}<%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*v<%(+,   0 QI1:9(    < 0C e-Theorem [Guan and Zhu, J. Graph Theory, 1999].0.  Bh o"`  C  'F N      B0   ;G0  6  B7.$G  6D  A)$G  6   W $G    6 @   A($G   6 & A$G   6  d  Cg&G   6 N   Cc&G  < s  For any outerplanar graph G,(0   8 G 1Gh2  s *"`BN B S '  N  S BJ h2  s *"` f h2  s *"`G 5 h2  s *"`:  h2  s *"`-9  h2  s *"` X N  S '; N  S  h2  s *"` ^ BN B S  h2  s *"`@ 9 N  S  B 9 h2  s *"`` 8 N  S   { N  S =h U& N  S f - N  B S  @ N ! S 5  h2 " s *"` ~ h2 # s *"`CR h2 $ s *"` F N % S ^  qN & S ~ p^N ' S c  N (B S  F & z 2 S '=  ,$D 0z 3 S BJ9 ,$D 0z 4 S B{  f ,$D 0z 5 S 'C { ,$D 0z 6 S # ,$D 0z 7 S 7 & ,$D 0 8 0- " / ,$ 0 :G  23 : <D1 2 JProof:(0H  0޽h ?_    ! % "#&"$'$(2345"6$#7 Z\l[]kfEoC;___PPT10.I+B#style.visibility<*2%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(+8+0+80 +B  0 ##<? (    <@ 0C e-Theorem [Guan and Zhu, J. Graph Theory, 1999].0.'F N      BD   ;G0  6XH  B7.$G  6 0 " d ,$  0 812 2 ? < 2 JProof:(0H  0޽h ?_  ! " # ' $%($&)&*+,-.$/&%0 33___PPT10.p+CD'  = @B D}' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(++0+20 ++0+30 ++0+40 ++0+50 ++0+60 ++0+70 ++0+80 ++0+90 ++0+:0 ++0+;0 ++0+=0 ++0+>0 +  0 um](   u 0ʑ " Z= ,$ 0 71 2 v 0Α " Z] @ ,$ 0 72 2f2 w 0" jzf2 x 0"   zF y S 6 6l f v f v,$D 0j@ f ? }f ?n2 z 0" z  ?N { S f N | S  z  ~ 0ԑ "  / v 73 2l z  z ,$D 0n2  0"`z P N  S 3z dN B S 3 z dH  0޽h ?_9wxy;wz{=zx|?wAx 33___PPT10._+tB=D ' ڑ = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*u%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*v%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D<' =%(D' =4@BBBB%()))D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(+p+0+u0 ++0+v0 +*  0 8VpB(    0( " Z=  71 2  0l " Z] @  72 2f2  0" jzf2  0"   zF  S 6 6F f v  f vxN f ?  f ?n2   0" z  ?N   S f N  S  z    0H "  / v 73 2l wfz 6 fwz 6,$D 0t2  6" A6@ wfz , wfz ,T B c $f~T  c $z   6 " wZ, 74 2]l  f3 Y f 3 Y,$D 0T  c $ f   6 " P 3 Y 75 2n2  0"  } "TB  c $D  ]l -+ L &+- L,$D 0T !B c $+ 2 " 6D " -L 76 2t2 # 6" =2N % S "QFl @0 C  /@0 C ,$D 0 ) 6 " { C  77 2v@ @0  .@0 T + c $ =TB , c $D @ Cn2 - 0"  )0 il  @P /  6@ P / ,$D 0 1 6 "   /  78 2T 3B c $ "9 ZB 4 s *D @" t2 5 6"  P Pz A}  F A} ,$D 0jN A}  G A} t2 H 68c"`jzt2 I 68c"`  zt2 J 68c"`z  ?T K c $8cn t2 L 68c"`A6T MB c $8cf~T N c $8c n t2 O 68c"` } "T PB c $8c3 *t2 Q 68c"`=2ZB R s *D8c @ Ct2 S 68c"` )0 ZB T s *D8c @" t2 U 68c"` P T V c $8c6 6H  0޽h ?     #! #%#+53HJKMIONJQPV 33___PPT10._+ΏDn'  = @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<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(+ L  0 7K/K  D(  )/8 ZF wZF  0  " =Z  71 2  0P " ] Z@  72 2n2  0" jzn2  0"   zN  S 6 6N f v  f vxN f ?  f ?n2   0" z  ?N   S f N   S  z    0P! "  / v 73 2N wfz 6   wfz 6n2  0" A6N wfz ,  wfz ,N B S f~N  S z   0& " wZ, 74 23N  f3 Y   f3 YN  S  f   0* " P 3 Y 75 2n2  0"  } "TB  c $D  -N -+ %  -+ %N B S + 2  0/ " -% 76 2n2  0" =2N  S "QFN @0    @0    0t4 " {   77 2~N @0   @0 N   S  =TB ! c $D @ Cn2 " 0"  )0 3N  @P   #  @P   $ 09 "     78 2N %B S  "9 TB & c $D @" n2 ' 0"  P t2 ) 6" JCT , c $Q - 6> " : 79 2TB . c $Dw"ZCn2 / 0"   N 0 S CQ N 1 S   2 0`C " w   810 2n2 3 0"   s N 4 S P^ N 5 S  /  6 00H "    811 2n2 7 0" s   : 0HL "  k  812 2TB ; c $D =s TB <B c $D=js n2 = 0"  S * N > S <  g n2 ? 0"   N @ S i  S N A S   B 0 R "  c S  814 2 F 0U "   813 2n2 G 0"  ~ V  H 0Z "  Q  815 2N K S   N L S  : n2 N 0" S)  N P S * )  R 0T^ "  F  816 2N S S ^? N T S 3_ =  U 0b "  H"  817 2n2 V 0" ) G n2 W 0"  wf N X S _  TB ZB c $DJ -  [ 0g " w   818 2n2 ] 0" f f  ^ 0k "  N + 819 2TB _ c $Dj f TB ` c $Dj f n2 a 0" ~ H  b 0,p " U  820 2TB c c $DP V HTB dB c $D  HTB e c $D  ) TB fB c $D~ n2 g 0" q<N h S   TB i c $D  j 0u " d 821 2N k S  0N l S `_ n2 m 0" t n 0z "  H 822 2N o S s PN p S R n2 q 0" q] r 0\ " 4 823 2N s S IN t S 3f 5n2 u 0" Im v 0胔 "  824 2dF  <   <t2  68c"` S * t2  68c"`  T  c $8ci  K t2  68c"` ~ V T  c $8c  T  c $8c : t2  68c"`S)  T  c $8c2 ! t2  68c"`~ H ZB B s *D8c  Ht2  68c"`q<T  c $8c  F "f m  "f mt2  68c"`JCZB  s *D8cw"ZCt2  68c"`  T  c $8cCQx t2  68c"`  s T  c $8cP^ t2  68c"`s  ZB B s *D8c=js T  c $8c^G t2  68c"`) G t2  68c"` wf ZB B s *D8cJ - t2  68c"`f f ZB  s *D8cj f T  c $8c`g t2  68c"`tT  c $8c{ Pt2  68c"`q]T  c $8c3n 5t2  68c"`ImF A}   A} jN A}   A} t2  68c"`jzt2  68c"`  zt2  68c"`z  ?T  c $8cn t2  68c"`A6T B c $8cf~T  c $8c n t2  68c"` } "T B c $8c3 *t2  68c"`=2ZB  s *D8c @ Ct2  68c"` )0 ZB  s *D8c @" t2  68c"` P T  c $8c6 6H  0޽h ?         '% ), /0/)13)43/5='>=@?=A?'KG'LN=P)SV3T 3WX#gdh$Vk%m3l&3o'qWp(qs)uWt56789?@ABCDEFGHI 33___PPT10i._+D='  = @B +^5  0 ##PI  (   F m   mlN  <    <t2   68c"` S * t2   68c"`  T   c $8ci  K t2   68c"` ~ V T   c $8c  T   c $8c : t2   68c"`S)  T   c $8c2 ! t2   68c"`~ H ZB  B s *D8c  Ht2   68c"`q<T   c $8c  N "f m   "f mt2   68c"`JCZB   s *D8cw"ZCt2   68c"`  T   c $8cCQx t2   68c"`  s T   c $8cP^ t2   68c"`s  ZB  B s *D8c=js T   c $8c^G t2   68c"`) G t2   68c"` wf ZB  B s *D8cJ - t2   68c"`f f ZB   s *D8cj f T   c $8c`g t2   68c"`tT   c $8c{ Pt2   68c"`q]T   c $8c3n 5t2   68c"`Im$N A}    A} jN A}    A} t2   68c"`jzt2   68c"`  zt2   68c"`z  ?T   c $8cn t2   68c"`A6T  B c $8cf~T   c $8c n t2   68c"` } "T  B c $8c3 *t2   68c"`=2ZB   s *D8c @ Ct2   68c"` )0 ZB   s *D8c @" t2   68c"` P T   c $8c6 62   0"`jz,$D 02   0"`Ji,$D 02   0"`z  ?,$D 02   0"`s ~ ,$D 0w   B( wT7,$ 0 KFor each uncolored vertex v, there are at most 3 colored neighbours in T. HL0HG: <z  m    m,$D 0t2   6|"`  T   c $|CQx t2   6|"` wf ZB  B s *D|J - T   c $|3n 5t2   6|"`Imt2   6|"`jzt2   6|"`z  ?T   c $|n T  B c $|3 *t2   6|"`=2`z     ~ ,$D 0t2   6|"`s  ZB  B s *D|=js H   0޽h ?O0  1  2  3   4   5  6   7  8   9  :   ;   < =   >   ? D  E   F   G    33C;___PPT10.}X+9Y@D'  = @B Dr' = @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<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(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+ 0 +PS  0 OOpH(  %/F ZF  ZF  0 " =Z  71 2  0<Ɣ " ] Z@  72 2n2  0" jzn2  0"   zN   S 6 6N f v   f vxN f ?   f ?n2   0" z  ?N   S f N  S  z   0˔ "  / v 73 2N wfz 6  wfz 6n2  0" A6N wfz ,  wfz ,N B S f~N  S z   0Д " wZ, 74 23N  f3 Y   f3 YN  S  f   0Ք " P 3 Y 75 2n2  0"  } "TB  c $D  -N -+ %  -+ %N B S + 2  0ٔ " -% 76 2n2  0" =2N  S "QFN @0     @0   ! 0`ޔ " {   77 2~N @0  " @0 N # S  =TB $ c $D @ Cn2 % 0"  )0 3N  @P   &  @P   ' 0\ "     78 2N (B S  "9 TB ) c $D @" n2 * 0"  P n2 + 0" JCN , S Q - 0x " : 79 2TB . c $Dw"ZCn2 / 0"   N 0 S CQ N 1 S   2 0p " w   810 2n2 3 0"   s N 4 S P^ N 5 S  /  6 0 "    811 2n2 7 0" s   8 0X "  k  812 2TB 9 c $D =s TB :B c $D=js n2 ; 0"  S * N < S <  g n2 = 0"   N > S i  S N ? S   @ 0 "  c S  814 2 A 0\ "   813 2n2 B 0"  ~ V  C 0 "  Q  815 2N D S   N E S  : n2 F 0" S)  N G S * )  H 0( "  F  816 2N I S ^? N J S 3_ =  K 0 "  H"  817 2n2 L 0" ) G n2 M 0"  wf N N S _  TB OB c $DJ -  P 0 " w   818 2n2 Q 0" f f  R 0 "  N + 819 2TB S c $Dj f TB T c $Dj f n2 U 0" ~ H  V 0D " U  820 2TB W c $DP V HTB XB c $D  HTB Y c $D  ) TB ZB c $D~ n2 [ 0" q<N \ S   TB ] c $D  ^ 0 " d 821 2N _ S  0N ` S `_ n2 a 0" t b 0\% "  H 822 2N c S s PN d S R n2 e 0" q] f 0) " 4 823 2N g S IN h S 3f 5n2 i 0" Im j 0. "  824 2F m k mlN  < l  <t2 m 68c"` S * t2 n 68c"`  T o c $8ci  K t2 p 68c"` ~ V T q c $8c  T r c $8c : t2 s 68c"`S)  T t c $8c2 ! t2 u 68c"`~ H ZB vB s *D8c  Ht2 w 68c"`q<T x c $8c  N "f m y "f mt2 z 68c"`JCZB { s *D8cw"ZCt2 | 68c"`  T } c $8cCQx t2 ~ 68c"`  s T  c $8cP^ t2  68c"`s  ZB B s *D8c=js T  c $8c^G t2  68c"`) G t2  68c"` wf ZB B s *D8cJ - t2  68c"`f f ZB  s *D8cj f T  c $8c`g t2  68c"`tT  c $8c{ Pt2  68c"`q]T  c $8c3n 5t2  68c"`Im$N A}   A} jN A}   A} t2  68c"`jzt2  68c"`  zt2  68c"`z  ?T  c $8cn t2  68c"`A6T B c $8cf~T  c $8c n t2  68c"` } "T B c $8c3 *t2  68c"`=2ZB  s *D8c @ Ct2  68c"` )0 ZB  s *D8c @" t2  68c"` P T  c $8c6 6f2  0"`jzf2  0"`Jif2  0"`z  ?f2  0"`i ~   <B  ',$ 0 ,For each uncolored vertex v, there are at most 6 colored neighbours in G . TN0IGC;  H  0޽h ?       # *( +, /0 /+13+43/5;*<;>=;?=*DB*EF;G+IL3J3MN[X\L_a3`3ceMdegiMh mo!nq"pr#smt$wvx%|}&~z'z(~)~*+,-./ 33___PPT10.ЍG+)DO'  = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+0 +tO  0 NN3H(  %/F ZF  ZF  0P " =Z  71 2  0T " ] Z@  72 2n2  0" jzn2  0"   zN  S 6 6N f v  f vxN f ?   f ?n2   0" z  ?N   S f N   S  z    0DZ "  / v 73 2N wfz 6  wfz 6n2  0" A6N wfz ,  wfz ,N B S f~N  S z   0_ " wZ, 74 23N  f3 Y   f3 YN  S  f   0c " P 3 Y 75 2n2  0"  } "TB  c $D  -N -+ %  -+ %N B S + 2  0h " -% 76 2n2  0" =2N  S "QFN @0    @0    0hm " {   77 2~N @0    @0 N ! S  =TB " c $D @ Cn2 # 0"  )0 3N  @P   $  @P   % 0 0 "  c S  814 2 ? 0t "   813 2n2 @ 0"  ~ V  A 0ؒ "  Q  815 2N B S   N C S  : n2 D 0" S)  N E S * )  F 0 "  F  816 2N G S ^? N H S 3_ =  I 0$ "  H"  817 2n2 J 0" ) G n2 K 0"  wf N L S _  TB MB c $DJ -  N 0 " w   818 2n2 O 0" f f  P 0죗 "  N + 819 2TB Q c $Dj f TB R c $Dj f n2 S 0" ~ H  T 0T " U  820 2TB U c $DP V HTB VB c $D  HTB W c $D  ) TB XB c $D~ n2 Y 0" q<N Z S   TB [ c $D  \ 0 " d 821 2N ] S  0N ^ S `_ n2 _ 0" t ` 0p "  H 822 2N a S s PN b S R n2 c 0" q] d 0 " 4 823 2N e S IN f S 3f 5n2 g 0" Im h 0 "  824 2F m i mlN  < j  <t2 k 68c"` S * t2 l 68c"`  T m c $8ci  K t2 n 68c"` ~ V T o c $8c  T p c $8c : t2 q 68c"`S)  T r c $8c2 ! t2 s 68c"`~ H ZB tB s *D8c  Ht2 u 68c"`q<T v c $8c  N "f m w "f mt2 x 68c"`JCZB y s *D8cw"ZCt2 z 68c"`  T { c $8cCQx t2 | 68c"`  s T } c $8cP^ t2 ~ 68c"`s  ZB B s *D8c=js T  c $8c^G t2  68c"`) G t2  68c"` wf ZB B s *D8cJ - t2  68c"`f f ZB  s *D8cj f T  c $8c`g t2  68c"`tT  c $8c{ Pt2  68c"`q]T  c $8c3n 5t2  68c"`Im$N A}   A} jN A}   A} t2  68c"`jzt2  68c"`  zt2  68c"`z  ?T  c $8cn t2  68c"`A6T B c $8cf~T  c $8c n t2  68c"` } "T B c $8c3 *t2  68c"`=2ZB  s *D8c @ Ct2  68c"` )0 ZB  s *D8c @" t2  68c"` P T  c $8c6 6f2  0"`jzf2  0"`Jif2  0"`z  ?f2  0"`i ~ :  <pϗ  ' LFor each uncolored vertex v, there are at most 6 colored neighbours in G. DM0IC; H  0޽h ?       ! (& )* -. -)/1)21-39(:9<;9=;(B@(CD9E)GJ1H1KLYVZJ]_1^1acKbcegKf km!lo"np#qkr$utv%z{&|x}'x(|)|*+,-./ 33___PPT10i.ЍG+D='  = @B +  0   & (    <ٗ 0C e-Theorem [Guan and Zhu, J. Graph Theory, 1999].0.'F N      Bޗ   ;G0  6  B7.$G  6  A)$G   6P   W $G    6 @   A($G   68 & A$G   6  d  Cg&G   6 N   Cc&G  < s  For any outerplanar graph G,(0  l      ,$D 0  <  Cw  $There is an outerplanar G such that (%0$ n  s *A ??   H  0޽h ? 33___PPT10n.pE +oDB'  = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+)8  0 $/<@(    0 Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N   @   B   ;G0  6  E3k+2.$G  6  A)$G  6%   W $G   6,) @   A($G  6- & A$G  62  d  Cg&G  67 N   Cc&G  0|< P & For any partial k-tree G,l0CCC  <D  JProof:(0l M  #,$D 0n2  0" Z}n2  0" M }n2  0" jN  S }N  S V} N  S Z99   0K " :,$ 0 71 2 ! 0O " [>,$ 0 72 2 " 0S " x%,$ 0 7k 2l ZP  (PM ,$D 0n2 $ 0"   TB % c $DZP 2TB & c $DM P0 TB ' c $Dj`  ) 0$Y "  ,$ 0 9k+1 2l jP  /P- ,$D 0t2 + 6" P  ZB , s *D 2P 2ZB - s *DM P~ ZB . s *Dj`P  0 08^ "  ,$  0 9k+2 2 z jP  1 P^ ,$D  0n2 2 0" P  TB 3 c $D 2P 2TB 4 c $DM P~ TB 5 c $Dj`P  6 0c " a 7,$  0 9k+3 2 z jP  7 PJ,$D  0n2 8 0" P  TB 9 c $D 2P 2TB : c $DM P~ TB ; c $Dj`P  < 0h " ,$  0 7n 2H  0޽h ??` 33___PPT10.j0p+ID'  = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =A@BBBB0B%(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<*)%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(++0+ 0 ++0+!0 ++0+"0 ++0+)0 ++0+00 ++0+60 ++0+<0 +5  0 **:f )(      0䅙 Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N    @    B4   ;G0   6\  E3k+2.$G   6<  A)$G   6   W $G    6̜ @   A($G   6 & A$G   6L  d  Cg&G   64 N   Cc&G   0 P & For any partial k-tree G,l0CCC   <l  JProof:(0   0D " Cc  71 2   0< " C* c  72 2l2 +  6"  t  /  0 " 3S  7n 2f2 0  0"  W f2 1  0"   f2 2  0"  \  f2 4  0"  #  f2 5  0"  q f2 7  0"  V f2 8  0"` Fs f2 9  0"` g f2 :  0"`  f2 ;  0"`  f2 <  0"`  2 =  0"` t ,$D 02 >  0"` p ,$D 02 ?  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRXv 2 @  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRu 2 A  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRg 2 B @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`  D  0hΙ " % 3k 22 E  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR/ v 2 F  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR  2 G  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR 2 H @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`E   I  0ә "  % 3k 22 O  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR } v 2 P  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR X  2 Q  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR B 2 R @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`H   S  0ؙ " P 3 % 3k 22 T  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRA v 2 U  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR^  2 V  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRP  2 W @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`   X  0ݙ " % 3k 22 _  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR'v 2 `  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRD 2 a  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR6v 2 b @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`|  c  0 " g% 3k 2F d  S  tW 2 e  6|"` t ,$D 02 f  6|"` q ,$D 0H   0޽h ?_`? @ A B E  F  G  H O P Q R T U V W _ ` a b != 0 d  33J B ___PPT10" .j0p+D '  = @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<*e %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*> %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*f %(+y/  0 #/K$k(  $ $ 0 Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N  $ @  $ B   ;G0 $ 6,  E3k+2.$G $ 6  A)$G $ 6    W $G  $ 6 @   A($G  $ 6@ & A$G  $ 6  d  Cg&G  $ 6, N   Cc&G  $ 0 P & For any partial k-tree G,l0CCC  $ <"  JProof:(0 $ 0' " Cc  71 2 $ 0`+ " C* c  72 2f2 $ 0"  t f2 $ 0"  W f2 $ 0"   f2 $ 0"  \  f2 $ 0"  #  f2 $ 0"  q f2 $ 0"  V f2 $ 0"` Fs f2 $ 0"` g f2 $ 0"`  f2 $ 0"`  f2 $ 0"`  f2 $ 0"` t f2 $ 0"` p f2 8$ 0"`  f2 9$ 0"` @ m f2 :$ 0"`  ;$  B CDEFAA5% Dk-q@  " ~   <$  BnCDEFAA5%n N@  " p0   =$  BCDEFAA5%^7@  "   >$  B CDEF5% Dk-q@  "   A$  B1CDEFAA5%1qN>q@  "  F B$ S  tW l2 C$ 6|"` t l2 D$ 6|"` p 0 E$  BCDEF|^7@  "` ,$D 02 F$ 6|"`  ,$D 00 G$  B1CDEF|1qN>q@  "` ,$D 02 H$ 6|"` W ,$D 0 I$ c $| tW ,$D 02 J$ 0"` a ,$D 02 K$ 0"` #  ,$D 0H $ 0޽h ?/@B$I$ 33___PPT10.j0p+[KD'  = @B De' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*K$%(+0  0 ?#7#6<("(  ( ( 00B Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N  ( @  ( BG   ;G0 ( 6K  E3k+2.$G ( 6O  A)$G ( 6T   W $G  ( 6Y @   A($G  ( 6] & A$G  ( 6b  d  Cg&G  ( 6g N   Cc&G  ( 0hl P & For any partial k-tree G,l0CCC  ( <t  JProof:(0 ( 0y " Cc  71 2 ( 0} " C* c  72 2f2 ( 0"  t f2 ( 0"  W f2 ( 0"   f2 ( 0"  \  f2 ( 0"  #  f2 ( 0"  q f2 ( 0"  V f2 ( 0"` Fs f2 ( 0"` g f2 ( 0"`  f2 ( 0"`  f2 ( 0"`  f2 ( 0"` t f2 ( 0"` p f2 ( 0"`  f2 ( 0"` @ m f2  ( 0"`  !(  B CDEF5% Dk-q@  " ~   "(  BnCDEF5%n N@  " p0   #(  BCDEF5%^7@  "   $(  B CDEF5% Dk-q@  "   %(  B1CDEF5%1qN>q@  "  F &( S  tW l2 '( 6|"` t l2 (( 6|"` p  )(  BCDEF|^7@  "` l2 *( 6|"`   +(  B1CDEF|1qN>q@  "` l2 ,( 6|"` W L -( c $| tW f2 .( 0"` a f2 /( 0"` #   5(  B C[DEFAA5% [-=q-[@  " % 0 P  6(  B C[DEF5% [-=q-[@  " %  2 7( 6|"` #  ,$D 00 8(  B C[DEF| [-=q-[@  "`% 0 P ,$D 02 :( 6|"` ]  ,$D 00 ;(  B C[DEF| [-=q-[@  "`%  ,$D 02 <( 0"`  ,$D 0H ( 0޽h ?/@&(-( 33 ~ ___PPT10^ .j0p+ղ6D2 '  = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<(%(+u$  0 KC1^,(  , , 0y&Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N  , @  , B~&  ;G0 , 6& E3k+2.$G , 6؆& A)$G , 6@&  W $G  , 6h&@   A($G  , 6(&& A$G  , 6& d  Cg&G  , 6&N   Cc&G  , 0T&P & For any partial k-tree G,l0CCC  , <X& JProof:(0f2 , 0"  ]  f2 , 0"` M z f2 , 0"`  f2 7, 0"  - f2 8, 0"  x f2 9, 0"  g f2 :, 0"`  f2 ;, 0"  #  f2 <, 0"   f2 =, 0"   f2 >, 0"    ?, 0&" PCp JUncolored vertex x 2CRB @,@ s *D`  S  C,  BCDEFAA5%aD>^D@  "   D,  BnCDEFAA5%nq6q@  "  f2 F, 0"`j  H, c $A ?? V- x dLB I, c $D g LB K, c $D w-  L,  BQCkDEFAA5%Qk k@  " :  M,  B C[DEFAA5% ->[[[D-@  "  :Zc  O,  BC[DEFAA5%[8-qH-[@  " % P  P,  BCDEFAA5%qq@  "   Q,  BnC=DEFAA5%={8n=@  " C &  S,  BCDEFAA5%MNO@  "   f2 T, 0"` ) f2 U, 0"`   Ll #   ^, #  ,$D 0n2 V, 0"`#  n2 W, 0"`  n2 X, 0"`  n2 Y, 0"`  l    ],  ,$D 0n2 Z, 0"` g n2 [, 0"` w n2 \, 0"`-  H , 0޽h ? 33___PPT10.j0p+)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<*^,%(+8  0 M"E"2;0!(  0 0 0 Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N  0 @  0 B   ;G0 0 6,  E3k+2.$G 0 6  A)$G 0 6t   W $G  0 6 @   A($G  0 6\ & A$G  0 6  d  Cg&G  0 6 N   Cc&G  0 0 P & For any partial k-tree G,l0CCC  0 <<ɜ  JProof:(0f2 0 0"  ]  f2 0 0"` M z f2 0 0"`  f2 0 0"  - f2 0 0"  x f2 0 0"  g f2 0 0"`  f2 0 0"  #  f2 0 0"   f2 0 0"   f2 0 0"    0 0Pќ " PCp JUncolored vertex x 2CRB 0@ s *D`  S  0  BCDEF5%aD>^D@  "   0  BnCDEF5%nq6q@  "  f2 0 0"`j  0 c $A ?? V- x dLB 0 c $D g LB  0 c $D w-  !0  BQCkDEF5%Qk k@  " :  "0  B C[DEF5% ->[[[D-@  "  :Zc  #0  BC[DEF5%[8-qH-[@  " % P  $0  BCDEF5%qq@  "   %0  BnC=DEF5%={8n=@  " C &  &0  BCDEF5%MNO@  "   f2 '0 0"` ) f2 (0 0"`   l2 *0 6"` #  2 20 6|"` #  ,$D 00 30  BC[DEF|[8-qH-[@  "`% P ,$D 02 40 6|"` ]  ,$D 00 50  BQCkDEF|Qk k@  "`: ,$D 02 60 6|"` g ,$D 02 70 0"`  ,$D 02 80 6|"`  ,$D 00 :0  BCDEF|qq@  "` ,$D  02 ;0 0"` j ,$D  0H 0 0޽h ? 33vn___PPT10N.j0p+ 9DD"'  = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*20%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*30%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*40%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*50%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*60%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*70%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*80%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;0%(+&  0 '  154(  4 4 0 Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N  4 @  4 B   ;G0 4 6  E3k+2.$G 4 6  A)$G 4 6@   W $G  4 6T @   A($G  4 64 & A$G  4 6P  d  Cg&G  4 6H N   Cc&G  4 0 P & For any partial k-tree G,l0CCC  4 <  JProof:(0f2 4 0"  ]  f2 4 0"` M z f2 4 0"`  f2 4 0"  - f2 4 0"  x f2 4 0"  g f2 4 0"`  f2 4 0"  #  f2 4 0"   f2 4 0"   f2 4 0"    4 0l( " PCp JUncolored vertex x 2CRB 4@ s *D`  S  4  BCDEF5%aD>^D@  "   4  BnCDEF5%nq6q@  "  f2 4 0"`j  4 c $A ?? V- x dLB 4 c $D g LB  4 c $D w-  !4  BQCkDEF5%Qk k@  " :  "4  B C[DEF5% ->[[[D-@  "  :Zc  #4  BC[DEF5%[8-qH-[@  " % P  $4  BCDEF5%qq@  "   %4  BnC=DEF5%={8n=@  " C &  &4  BCDEF5%MNO@  "   f2 '4 0"` ) f2 (4 0"`   f2 )4 0"` #  l2 *4 6|"` #   +4  BC[DEF|[8-qH-[@  "`% P l2 ,4 6|"` ]   -4  BQCkDEF|Qk k@  "`: l2 .4 6|"` g f2 /4 0"`  * 34 s BCDEFA5%-N=x<@  "` z ,$D 0l2 54 6|"`  H 4 0޽h ? 33___PPT10.j0p+/D'  = @B Dw' = @BA?%,( < +O%,( < +Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*34%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*34%(+:>  0 &&08<8&(  8 8 06 Z7 V"Theorem [Zhu, Discrete Math.,2000]##C*F N  8 @  8 B8<   ;G0 8 6`@  E3k+2.$G 8 6@D  A)$G 8 6I   W $G  8 6M @   A($G  8 6R & A$G  8 6PW  d  Cg&G  8 68\ N   Cc&G  8 0 a P & For any partial k-tree G,l0CCC  8 <pi  JProof:(0f2 8 0"  ]  f2 8 0"` M z f2 8 0"`  f2 8 0"  - f2 8 0"  x f2 8 0"  g f2 8 0"`  f2 8 0"  #  f2 8 0"   f2 8 0"   f2 8 0"    8 0q " PCp JUncolored vertex x 2CRB 8@ s *D`  S  8  BCDEF5%aD>^D@  "   8  BnCDEF5%nq6q@  "  f2 8 0"`j  8 c $A ?? V- x dLB 8 c $D g LB  8 c $D w-  !8  BQCkDEF5%Qk k@  " :  "8  B C[DEF5% ->[[[D-@  "  :Zc  #8  BC[DEF5%[8-qH-[@  " % P  $8  BCDEF5%qq@  "   %8  BnC=DEF5%={8n=@  " C &  &8  BCDEF5%MNO@  "   f2 '8 0"` ) f2 (8 0"`   f2 )8 0"` #  l2 *8 6|"` #   +8  BC[DEF|[8-qH-[@  "`% P l2 ,8 6|"` ]   -8  BQCkDEF|Qk k@  "`: l2 .8 6|"` g f2 /8 0"`   28  BCDEFAA5%H{M@  "   2 38 6|"`  ,$D 00 48  BCDEF|H{M@  "`  ,$D 0~ 68 0d " ) p ,$  0 6There are at most k +2(k+1) colored neighbours of x. h7 2CG3CGCC% B 78 s *DԔ = ,$D  0 88 B "  i 5,$  0 @ k +2k+1=3k+1 C2 :8 0"`  ,$D 02 ;8 6|"`  ,$D 0* <8 s BCDEFA|il6[@  "` Z& ,$D 0H 8 0޽h ? 33___PPT10.j0p+̮ D-'  = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*38%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*48%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<8%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*68%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*68D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*68D'' =%(D' =%(Dw' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*78%(D' =-s6Bwipe(down)*<3<*78DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*88%(+p+0+680 ++0+880 +@*  0 &%.(  `2  s *"```2  s *"`R`f2  0"`j2  <ܜ <7\ Ft colors" 0 Gf2  0"`z 2f2  0"` 2`2  s *"` _`2  s *"`$_2  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR B f2  0"` 2f2  0"`S2f2  0"`622  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR@  2  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR 2 @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`m@   s *A W @??/ ;.x @d2  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRB2  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRA_2 @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`n  s *A W A??0/x Ad2  CRENGqJRQ `TZL`TR`TZL`TRRL`TRR2  CRENGqJRQ `TZL`TR`TZL`TRRL`TRRA1y2 @ CRENGqJRQ  `TZL`TR`TZL`TRRL`TRR"`n   s *A W B??0Q/x Bd2 ! CRENGqJRQ `TZL`TR`TZL`TRRL`TRR?2 " CRENGqJRQ `TZL`TR`TZL`TRRL`TRR I2 $ 0"`_,$D 02 % 0"`J_,$D 02 & 0"` _,$D 02 ' 0"`)_,$D 04l j   *V Tv ,$D 0 ( Bԫ j   b0game chromatic number game coloring number1 1 ) c $A C?? x CdA3$l T 5 .  ,$D 0 , H\+ T A + Fgame coloring number  - s *A ?? 5x d 3H  0޽h ?    ! " 33___PPT10.V?+*YDn'  = @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<*%%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(+#-   0 :,2,AA+(  1 <w  # #"."rw    <x  ?<w5 eGame chromatic numberC @`  <ʞ  ?Hw  R17C @`  <Ԟ   ?"`9 H  YPlanar  (C @`  <pݞ  ?9  UZhu (C @`  <ȹ  ?Hw T3k+2C @`  <   ?"`9 H ^Partial k-treeC @`   <  ?9  SZhuC @`   <  ?H w T3k+1C @`   <   ?"`9 H _Interval graphsC @`   <T  ? 9   Faigle, Kern, Kierstead, Trotter!!C$   @`   <x  ?H'w  Q7C @`  <)   ?"`9 'H  u Outerplanar  C  @`  <1  ?'9  \ Guan and Zhu  C @`  <:  ?H.w' Q4C @`  <DD   ?"`9 .H' WForestsC @`  <F  ?.9 '  Faigle, Kern, Kierstead, Trotter!!C$   @`  <X  ?H5w. [ Upper bound  C @`  <Hb  ?9 5H. UGraphC @`  <S  ?59 . VAuthorC @`fB  6o ?<w<`B  01 ?.w.`B  01 ?'w'`B  01 ? w `B  01 ?w`B  01 ?wfB  6o ? w fB  6o ?< fB  6o ?w<w ZB  s *1 ?5w5`B   01 ?9 59 `B ! 01 ?H5H Y  "# #"."w   # <x  ?  dGame coloring numberC @` $ <8  ? R17C @` % <s  ?  V11"G3 @` & <  ?  YPlanar  (C @` ' <D  ?  T3k+2C @` ( <L  ?  X3k+2"G3 @` ) <  ?  ^Partial k-treeC @` * <  ?   T3k+1C @` + <  ?   T3k+1C @` , <ʢ  ?  _Interval graphsC @` - <0Ԣ  ?   Q7C @` . <`ݢ  ?   Q7C @` / <آ  ?  u outerplanar  C  @` 0 <  ?   Q4C @` 1 <`  ?   Q4C @` 2 <  ?  WForestsC @` 3 <  ?   [ Upper bound  C @` 4 <  ?   [ Lower bound  C @` 5 <  ?  UGraphC @`fB 6 6o ?`B 7 01 ?  `B 8 01 ?  `B 9 01 ?  `B : 01 ?  `B ; 01 ?fB < 6o ?fB = 6o ?fB > 6o ?ZB ? s *1 ?  `B @ 01 ? `B A 01 ? H  0޽h ? 33___PPT10i.jLp+D='  = @B +/   0 F> (    nP fjJ?The Ende0}fԚ" H  0޽h ? Z\l[]kfEo___PPT10i.IRs3D )eK +os1;p}پã! Smշg;$'睌3٤{Tvuܧoc˹`yM$nZ>>kS!aB0@DĖVraȥZ=~W% <SxXOAf]*F+1  (c8x1@ m^Rb'Gn?Acb⁛fv)tA6}7{ߣ0{*T[.|; ӕ.> ?>beP=t-kl Nuz4.` M, ΊfQ{܅ L!D+>hvݕWKٌj*pF1N0knB싸7s|T,|_)ڿ*ԾNlb;?;{8Y+:Ww"? ļb}n-M bwb'׹Cn;:9 _aC?{C Ϲ#&`+ZrN!EIZYj*9\u#hTg83$=rǠaCO(y8+m㷖IH4bxf>beR&w9>ax3Gm]nѿ5L޿,i.4إ~-z@| NjG7   R)T#kovPm6S6%[52/wfM}v"BfCtZ5kӝDTe1T/ywd<NZ]Sd$[+ԙB$c䓉 @cqUC:)̂aX =[ 5ңs*`uS ƚZxXOQ.E$a@ A j Sb m)-4548yBLyo']YZR (J۠dc\*E-Nɤ _ <8$ST ř3|K"C}<6yF"+'X^1ͤg[^.s"0>|Ckh9JÊ0g.t`ސŸA}kLE1d92ˠP;e?>>/%e!owyt(N&QPRЙS@|# ˍ0Uz 9R{QhS!z^FzZZߩ?aYKIbW'ZcsxqLDc T8wtYg/bX}} SQfNiŁ+hx$y"X :m:txO>pƻsRv}k'j]VSTɧ]Տf+0zO-kt>㜚d1_ʟ(䷟КST8;eNKKI|Hb`$73leqvLuR@UXo|3#N`IC/vu(1\Є+pxhtoԅݪϵYқ2ҝzf7o?g3FX ^YF+ +ࢾpi6_!< p|r<=^l?l]&1$'\sƇO 3 _fNxIP[ew3xt8 t䇵W9T,b .xJĻ3马10&)# %j3ӣRaغ|8|~zK-1:w74ǠW) | +ߛ=(yK4o7w<(;Z͢RxD q!IA&~^Art"+qC}0qwt'܍%goAK_c۵w[zJ,@!O*ӑUOn0(n. )G?f1`ʕnvJǂ~xWkA3V3 xXh6M="(I(M5!|4ְ7A œ< ƃՋ=7;qMMB62ɛf޼ͫ;{0Vju7 ڈ}`V벿`S! yt 9_nj }lexQo3~o Mf ?awPքRTR"Y*@|VHcVT1$Tz)c<9c㠐^+-. ޹t6Q"g9xssEӾYro6}§NG9]g16sچz[=L9 C֏ MrݦҐ3/8Ą&)+|1 3z" /Ϝ-"cGZG0nZr=N_^I] z:ӦA5aJ`{Xh#|=k+OF!siT ! c:в+S,чH:G>?#{nmKF{bR6mg_w(+󿟐LxXAkQ6V ɆAQ4Tffk؛ /Ѓ'GEhIP Jg޾ KȮJj0;73ovۼ<ɳ <(EacH$I^KAM0VBr`P88Y//ȜnqAV1G6nhƦٹQM<נ ŰfgaG: K~VюZz/%rEh@F#) =' 7 6Pu(ί>a0dn]c3A%Y?qJ(Й3HQ9&&0 T^:&iޜm1EUoNim--7Ǟ;Lo22 gIʖ6+REr(B,)y8o3h:1ufdÏQhc Rił]^rV1e^׵)R,6tsYߚ$]4R7=qJ]f);;$/+V 'm<&'$'s?-tnWYlG^R';"$o0SyaMȑ\:b]|`pMʐc\4zmYEQe erkjX%[ vŪ 8]ye{8>D/xXAkA~3XzXsdCj (C$ţ4Dll {P<Dz!^ AP Jߛ KjeW%5_x3og{{yzg0ӠA)!qՉ쒼^'ԟ`n#&6dŸ &8é8Y+Ϝq~T1G6Wn踏/x>ht`TSh.%@!YG_Ucno~(\2zMHCd!W~wN:Aw~A @Fnڕ&FW߸=wQbT@r)?߃c"> ; @5C2~itq6|}~saK̟a$iB۬xJVQEQh%Y\"qQa2udj/ȚKFa9& yb:K ݿk}R !r43{E2o^昻',y89x郱<&#('s^}I h , K/“G1牜Ӵ UȒ\:{b]~4`xM9's6sQfMq3tv<մU3)q;V|?!0_#~^#xWkA36ͤbL[ZDPLFQx6 MiI-9QYP^l,Jŋl؄X7aMe{>'럇6+C΁Ms KU*M)HU]ū*D e(6®!kKw7GݺQLSWa-4CgﱲƃtH Sh !*Gw,i!iB .4,fa:QV&]4 !A-NO\w_s;}ϊN C>9׽Pϝ{0z"cw6װ' em_]ւv#n 3aTUSnpz"qR s{̏0&{5i%AomIo^>ʗ/VLf6<"tlO~Tug5!}'{O0ҾyQH)BDJ]=~q>+ 4G'6 .iL=b "N#3g5[Rٌ NQ.jsA0?OOrDrN~_NہV'779hpT8p@w5 cߛg`6b@a&Vy-t G7#m.xXAOA~3ۂ] J x$(8zh5Z0xxx7ы!P ă'cA#7f)c`7;^g{ͶiPFWAݒjlm +%vK5iȥ*+4^Y0`R|9p[{NxB oodn~yQ^r)L ďN:pFAB` 3O:p"R[t z:߀<,eofen;J:!ٕ\}@{N.)kJqny@)ԩWcd:QPHS< j&z |*$. q@ȼS_!tW;C>?g3FX ^YF+ +ࢾpi6_!< p|r<=^l?l]&1$'\sƇO 3 _fNxIP[ew3xt8 t䇵W9T,b .xJĻ3马10&)# %j3ӣRaغ|8|~zK-1:w74ǠW) | +ߛ=(yK4o7w<(;Z͢RxD q!IA&~^Art"+qC}0qwt'܍%goAK_c۵w[zJ,@!O*ӑUOn0(n. )G?f1`ʕnvJǂ~.xXAOA~3ۂ] J x$(8zh5Z0xxx7ы!P ă'cA#7f)c`7;^g{ͶiPFWAݒjlm +%vK5iȥ*+4^Y0`R|9p[{NxB oodn~yQ^r)L ďN:pFAB` 3O:p"R[t z:߀<,eofen;J:!ٕ\}@{N.)kJqny@)ԩWcd:QPHS< j&z |*$. q@ȼS_!tW;C>?g3FX ^YF+ +ࢾpi6_!< p|r<=^l?l]&1$'\sƇO 3 _fNxIP[ew3xt8 t䇵W9T,b .xJĻ3马10&)# %j3ӣRaغ|8|~zK-1:w74ǠW) | +ߛ=(yK4o7w<(;Z͢RxD q!IA&~^Art"+qC}0qwt'܍%goAK_c۵w[zJ,@!O*ӑUOn0(n. )G?f1`ʕnvJǂ~.xXAOA~3ۂ] J x$(8zh5Z0xxx7ы!P ă'cA#7f)c`7;^g{ͶiPFWAݒjlm +%vK5iȥ*+4^Y0`R|9p[{NxB oodn~yQ^r)L ďN:pFAB` 3O:p"R[t z:߀<,eofen;J:!ٕ\}@{N.)kJqny@)ԩWcd:QPHS< j&z |*$. q@ȼS_!tW;C>?g3FX ^YF+ +ࢾpi6_!< p|r<=^l?l]&1$'\sƇO 3 _fNxIP[ew3xt8 t䇵W9T,b .xJĻ3马10&)# %j3ӣRaغ|8|~zK-1:w74ǠW) | +ߛ=(yK4o7w<(;Z͢RxD q!IA&~^Art"+qC}0qwt'܍%goAK_c۵w[zJ,@!O*ӑUOn0(n. )G?f1`ʕnvJǂ~xYMLA~3 J+1V"AQ &R p1Ђ-h+g.ƫ11&$&^@O[Ζ MC,kfy3~vm' i BJ >ޖLR$* QaqHMaKܟeDZQY:TFxBP8I7GHүɋuEĴc4!'ʱޞޒל!Dx :΃nqeqi C3_v,_bq^J$0xR*~]#r7cVK=N"׺([^yOx4O(mK0,bK2^?;A?/ouƢ16Ꮗ*oXa; rɃ_i]xɧcjZL38g0µ0;•3_ϥ wpug^]ԡ/N~ y7sC'o3cNb&Bڝ....璟a ↱\di^D KL+ FӃxXkAffSh$C=Y/R4J[ɵQ5Iۤ8 xgx*xL/Qћfv&DZ7!J[^{3oޛ6oJЀ0PcD\7\TDAT0 ]9X#MU]aG]s!Eҽ#/Yv?pt:a3z܌P|+XCV$;պjVBPTΦI{LT"A;./)/J?ѿLųV$2X~oYַDT(sqȥ-oܚ<&*wMK+L5{S>l]2g&w \<+aܭSsu1]:ȩYwQ_6L AM^Zd3^Y咙5>:Fҍq|?tŝ~a/B]<滠 d[30- %yx6M Cn;_MxNh۶C ߤXw_ <xXkSAZB Ă/h R-xmcM4!xAz => W8r yVy^= ^ ?bӭmm~yb \- R 4,Vipӷ0hb>楞ZTw.o֍XtqGe$3d9 *:i =`/^6ӳfNav:O[BX-`[F;Ξ9;B*1w gSU-4EՋRĘ. 6 sґaN c=FI3 *(m1,X;[VKG#u@"ɦN"Uqd_9ı pm]˪犖^ߛ<{SU8[V3t+lxk@HybZ,a|#Ca0]]\:8Ћh A|ZOx,e9g*6I\\l2~(xտD棜xX;hSak57Abn B(Fm `I&m.I]g: xW){K仜|8E|8 fmu$l!2uf)f=LD9C E~ vE@EPF/ > zO^TNqt2n#t-ta/ Xy\ǭ_@ 1 /gQzcԬ ~9MҼ79IXE=?~P865_^k~;^w_2~ 2>K?b]ǃ1tMû6?9K{3p a斀gmL~#^ccmJۘjYT}Q $pߐ8'<e0&(6{δłf3՛ egJ?yDlKgk\)ӟm&UKM"-ZT#ڷ-4vqVQ {81_TLIFCs{vZ٫ϧ/A}~82891X)UJ*fRћH[ul!9g^3P_}?xXkAښMiXb/h R-1Q5Il KEIċw={A/~@)^ofgBHQvK䷼웯}Ɠgt4,4[! w0(pH5[Z05{PDžJ^G;qiA/sxe-yof˧Wr.&rA݆ AAf:5U?e4ϘS^B|UL?&7&Ǜ\ ipBȲJҺt˗W:lߎ>D?>.#_2;v[a2yz6q[DmSQBU`l!]6߈dg=6Z,(P\u]/"GĘ;>`"sڅ_{ Ͻw ruR[(4 p~]3uyu%1@p-~.xXAOA~3ۂ] J x$(8zh5Z0xxx7ы!P ă'cA#7f)c`7;^g{ͶiPFWAݒjlm +%vK5iȥ*+4^Y0`R|9p[{NxB oodn~yQ^r)L ďN:pFAB` 3O:p"R[t z:߀<,eofen;J:!ٕ\}@{N.)kJqny@)ԩWcd:QPHS< j&z |*$. q@ȼS_!tW;C>?g3FX ^YF+ +ࢾpi6_!< p|r<=^l?l]&1$'\sƇO 3 _fNxIP[ew3xt8 t䇵W9T,b .xJĻ3马10&)# %j3ӣRaغ|8|~zK-1:w74ǠW) | +ߛ=(yK4o7w<(;Z͢RxD q!IA&~^Art"+qC}0qwt'܍%goAK_c۵w[zJ,@!O*ӑUOn0(n. )G?f1`ʕnvJǂ~xXMhA~3FMMa=HmCSi=LT+Mڴin )x"xЂSZ/Ά&쾙̼י{_:2y7` A*] NN@KimYTaD@f)0TCIgB-ƹ߮x1ɖwj(2pFuYYЅBo~ϡ'zC*#xV&t{opgyfa;8TtBG=݅}MvAe.H>Q5lpMauW2u"ɰΩMD{9YL34s|v=wE@;ܫ{#3G㭇?\+ժ.1&2㧟HA,,m> cԽu, -mz]yum&|vwef%rwv*Z~]3Ss.Md_c5t,5<,na3O*;oFgJ/.JFgX3p|0g=*lOEOqnJ]-mxK+xXAhA3i).RK"xh)V=HD#6J[n0$m<"QTeAg+0fE60"Oc0S7O2臸,4+8sǽPKyq}tOls҇xz#_ RaPKS I'jF1+?n׀w_9u}ԙCmkw~e߄;ڪ?۬#~)ߗ? PPkd,oaxrQEtVWB8zX k3,tԎ5vyf`yU*@I|֍I-Ήm̧Gc4<9bkӣZrޯ9gZo4I+4|24(ctpl0qG-__h%gӟ>eJXlE\, gBO:rj93C%?*~` ^8(6sFܜ6Dfz)DO+[>yt;?X7 -@EE]%D@M  ]7P(tX-"9_ 6Y?rY1Oh+'0T hp    v 1wujjwujj268Microsoft Office PowerPoint@`\@ ZI@1DGSg  )'    """)))UUUMMMBBB999|PP3f333f3333f3ffffff3f̙3ff333f333333333f33333333f33f3ff3f3f3f3333f33̙33333f333333f3333f3ffffff3f33ff3f3f3f3fff3ffffffffff3ffff̙fff3fffff3fff333f3f3ff3ff33f̙̙3̙ff̙̙̙3f̙3f333f3333f3ffffff3f̙3f3f3f333f3333f3ffffff3f̙3f3ffffffffff!___www4'A x(xKʦ """)))UUUMMMBBB999|PP3f3333f333ff3fffff3f3f̙f3333f3333333333f3333333f3f33ff3f3f3f3333f3333333f3̙33333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffff3fffffff3f̙ffff3ff333f3ff33fff33f3ff̙3f3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3ffffffffff!___wwwýýýýýýýüýüýüýüýüýýýüýüýüýýýüýýýýýýýüýüýüýüýüýýýüýüýüýýýüýýýýýýýüýüýüýüýüýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýüýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýüýýýýýýýýýýýüýýüýüýýüýüýýýýüýüýýýüýýýüýýüýüýýüýüýýýýüýüýýýüýýýüýýüýüýýüýüýýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýüýýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýýüýýýýýýýýýýýýýýýýýýüýýýýýýýýýüýýýýýýýýýýýýýýýýýüüýýýýýýýýýýýüýüýüýýýýýýýüýýýýýýýýüýýýýýýüýýýýýýýýýýýüýüýüýýýýýýýýýýýýýýýüýýýýýýýýýýüýýýýýýýýýüýýýýýýýýýýüýýýýýýýýýüýýýýýýýýýýüýýýýýýüýýýüýüýüýýýüýýýýýýýüýüýüýýýüýýýüýüýüýýýüýýýýýýýüýüýüýýýüýýýüýüýüýýýüýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýüýýýýýýýýüýýýüýýýýýýýüýüýýýüýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýüýýýýýýýýýýýýýýüýýýýýýýýýüýüýýüýýýüýýüýüýýüýüýýýýüýüýýüýýýüýýüýüýýüýüýýýýüýüýýüýýýüýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýüýýýýýýýüýýýýüýüýýýýýüýýýýüýýýýýýýüýýýýüýüýýýýýüýýýýüýýýýýýýüýýýýüýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýüýýýýýýýýýüýýýýýýýýüýýýýýýýýýüýýýýýýýýüýýýýüýýýýýýýüýüýüýüýüýýýüýüýüýýýüýýýýýýýüýüýüýüýüýýýüýüýüýýýüýýýýýýýüýüýüýüýüýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýüýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýüýýüýüýýüýýýýüýüýýýýýýüýýüýüýüýüýýýýüýýýýüýýýüýýüýüýýüýýýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýüýýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýýüýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýüüýýýýýýýýýýýüýüýüýýýýýýýüýýýýýýýýýýýüýüýüýýýýýýýüýýýýýýýýýýýüýüýüýýýýýýýýýýýýýýýýüýýýýýýýýýýüýýýýýýýýýüýýýýýýýýýýüýýýýýýýýýüýýýýýýýýýýüýýýýýýüýýýüýüýüýýýüýýýýýýýüýüýüýýýüýýýüýüýüýýýüýýýýýýýüýüýüýýýüýýýüýüýüýýýüýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýüýýýýýýýýüýýýüýýýýýýýüýüýýýüýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýüýýýýýüýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýüýýýýýýýýýüýüýýüýýýüýýüýüýýüýüýýýýüýüýýüýýýüýýüýüýýüýüýýýýüýüýýüýýýüýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýýýýüýýýýýýýüýýýýüýüýýýýýüýýýýüýýýýýýýüýýýýüýüýýýýýüýýýýüýýýýýýýüýýýýüýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýüýýýýýýýýýýüýüýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýüýýýýýýýýüýýýýýýýýýüýýýýýýýýüýýýýýýýýýüýýýýýýýýüýýýýüýýýýýýýüýüýüýüýüýýýüýüýüýýýüýýýýýýýüýüýüýüýüýýýüýüýüýýýüýýýýýýýüýüýüýüýüýýýüýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýüýüýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýüýýýüýýýýýýýüýüýýýýýýýýýýýýüýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýý՜.+,0     pùjpnsysuV|% -Arial sөTahomaзTimes New RomanSymbol w]²]pMicrosoft {s边 3.0 Game chromatic number of graphs v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 112-tree v 13 v 14 v 15 v 16 v 17 v 18 v 19 v 20 v 21 v 22 v 23 v 24 v 25 v 26 v 27 v 28 v 29 v 30 v 31 v 32 v 33 v 34 v 35 v 36 v 37  ϥΦr ²]pdO OLE A{ vD%_wujjwujj  !"#$%&'()*+,-./0123456789:;=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesvCurrent UserSummaryInformation(TPowerPoint Document(<DocumentSummaryInformation8