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P(Iz   RF f3+/.aqLiHjjjGGG      ^n2$&g+yc_02$?k~j02$uU(0zg5BGV%M$$$$$$$$$$$2$}|n1 r0$2$m:l8MT>?M< 0AAP'/f@8w]Bʚ;ʚ;g4KdKd0pppp@ <4dddd 0,3 <4!d!d. 0,80___PPT10 pp?  %Z-JOn the Existence of Multicolored Parallelisms of Isomorphic Spanning TreesKKXHung-Lin Fu and Y. H. Lo Department of Applied Mathematics NCTU, Hsin Chu, Taiwan 30050 &YX"A Preliminaries A proper k-edge coloring of a graph G is a mapping from E(G) into {1,& ,k} such that incident edges of G receive distinct colors.t 9=Facts on Edge-ColoringsNLet G be a simple graph with maximum degree D(G). Then, the minimum number of colors needed to properly color G, c(G), is either D(G) or D(G) + 1. (Vizing s Theorem) G is of class one if c(G) = D(G) and class two otherwise. Kn is of class one if and only if n is even. Kn,n is of class one. T(,,FF*> Rainbow SubgraphV/ vLet G be an edge-colored graph. Then a subgraph whose edges are of distinct colors is called a rainbow subgraph of G. It is also known as a heterochromatic subgraph or a multicolored subgraph. Note that the edge-coloring of the edge-colored graph may not be proper. In this talk, all edge-colorings are proper edge-colorings. Therefore, a rainbow star can be found easily.^wZ`; LnP( 7 ?!Rainbow 1-factorTheorem(Woolbright and Fu, JCD 1998) In a (2m-1)-edge-colored K2m where m > 2, there exists a rainbow 1-factor. (The coloring is arbitrarily given.) Conjecture(Transversal) In an n-edge-colored Kn,n, there exists a rainbow 1-factor for each odd integer n. (The coloring is arbitrarily given.) Note: There exist n-edge-colorings of Kn,n such that n edge-disjoint rainbow 1-factors exist. Z "##   :#) 4> 7@"Multicolored Subgraph  Given an (n-1)-edge-colored Kn, a multicolored Hamiltonian path exists for n > 4.(!?) How about multicolored spanning trees? x/*aA#Orthogonal Latin SquaresdA Latin square of order n corresponds to an n-edge-coloring of Kn,n. A Latin square of order n with an orthogonal mate provides n edge-disjoint multicolored 1-factors of Kn,n. 1 2 3 1 2 3 2 3 1 3 1 2 3 1 2 2 3 1 nZZ@ h #   >?iC% Room SquaresA Room square of side 2m-1 provides a (2m-1)-edge-coloring of K2m such that 2m-1 edge- disjoint multicolored 1-factors exist. 35 17 28 46 26 48 15 37 13 57 68 24 47 16 38 25 58 23 14 67 12 78 56 34 36 45 27 18 ~ZZZ? Z f /B$Multicolored Tree ParallelismK2m admits a multicolored tree parallelism (MTP) if there exists a proper (2m-1)-edge-coloring of K2m for which all edges can be partitioned into m isomorphic multicolored spanning trees. 2/ D&K6 admits an MTP$ T1 T2 T3 Color 1: 35 46 12 Color 2: 24 15 36 Color 3: 25 34 16 Color 4: 26 13 45 Color 5: 14 23 56       Three Conjectures on MTP Constantine s Weak Conjecture For any natural number m>2, there exists a (2m-1)-edge-coloring of K2m for which K2m can be decomposed into m multicolored isomorphic spanning trees. (MTP) *) Brualdi-Hollongsworth Conjecture!!$  If m>2, then in any proper edge coloring of K2m with 2m-1 colors, all edges can be partitioned into m multicolored spanning trees. P3i>Constantine s Strong Conjecture $ If m>2, then in any proper edge coloring of K2m with 2m-1 colors, all edges can be partitioned into m isomorphic multicolored spanning trees. X28 0 Theorem (Constanine, 2002)     If m 1 or 3 and K2m admits an MTP, then for all positive integer r > 1, admits an MTP. Lemma K12 admits an MTP. Thanks, Peter Cameron, Gap software works! There exists a proper edge coloring of K6,6 with 6 colors in which all edges can be partitioned into 6 multicolored subgraphs isomorphic to K1,33K2 2K1.CZ  0"( ,( b "  $Theorem (Akbari, Alipour, Fu and Lo) % ,  For m is an odd positive integer, then K2m admits an MTP. Corollary. Constantine s Weak Conjecture is true.:q+ )Proof of the Main ResultAn m-total-coloring of a graph G is a mapping from V(G)E(G) into {1,2,3, & , m} such that incident vertices and edges receive distinct images.D$W$ An example: m = 5(Z %An example: m = 5Z H'An example: m = 5Z I(An example: m = 5Z *An example: m = 5 -An example: m = 5 1An example: m = 5Z 2An example: m = 5 3An example: m = 5 4An example: m = 5Z J)Concluding RemarkIf the edge-coloring is arbitrarily given, then the proof of finding MTP is going to be very difficult. We may drop  isomorphism for the above case and find m multicolored spanning trees of K2m. (k g ^g)0  ` 33PP` 13` 3333` Q_{` 333fpKNāvI` j@v۩ῑ΂H>?" dd@,?n<d@ `7 `2@`7``2 n?" dd@   @@``PR    @ ` ` p>>     (     <n" 4 0C  THrd" 4 0C  <v"U_ 4 0C  Txd">& 4 0C  N{"P 4 0C  <4"p 4 0C  C x0?d?"bUv 4 0C  <P #" `   L cN NN}/kGrjL#j_    0` "   |0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\    6܌ "]}  b*0  6̐ "] }   d* 0  6d "]T}  d* 0b  <'d޽h ?"` 333380___PPT10.(г%  Blends   P H @$ (  $T + $ "+bb P@ $# "Dwoh $ s *"PP $ Bd" P@bb P 0  $# "Nyh $ s *"P   $ Bd"P 0 z  $ <" a*h  $ s *"  $  f?d?"+)  $ B ?#" ` p  L cN NN}/kGrjL#j_    $ 0 " `    N cN NN}/kGroRjL#j_  $ 6 "`p   f*"0 $ 6 "`p   h*$0 $ 6x "`  h*$0b $ <'d޽h ?"` 333380___PPT10.(г% 00(  x  c $ $  x  c $ $ `    H  0޽h ? 33___PPT10i.v Q+D=' | = @B +W   P8L(  8r 8 S `d     r 8 S   6   fl   8 ,$D  0`2 8 0 0 `2 8 0m ( `2 8 0 ( `2 8 0L f z `2 8 0 f ZB  8B s *D  ZB  8 s *D) V f ZB  8 s *D )  f ZB 8 s *Dz ZB 8 s *D0 m  8 0  00 $ RA 3-edge coloring of 5-cycle 2H 8 0޽h ? 3333C;___PPT10.v k+ j D' | = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-m6Bbox(in)*<3<*8+  `$(  r  S      r  S 0    H  0޽h ? 3333  p$(  r  S 8     r  S     H  0޽h ? 3333  $(  r  S      r  S p   H  0޽h ? 3333  vn(  r  S       r  S        C AC:\Program Files\Common Files\Microsoft Shared\Clipart\cagcat50\BD06639_.WMF0 H  0޽h ? 3333  $(  r  S      r  S     H  0޽h ? 3333   $(   r   S `     r   S     H   0޽h ? 3333  $(  r  S d     r  S (    H  0޽h ? 3333:   z(  r  S x      r  S 4!    N8 `@    P`  fB  6Do` ` fB  6Do@@@fB  6Do    ZB  s *Do`@@ ZB  s *Do@@  H  0޽h ? 3333  `:(  `r ` S '      ` S 9  f   "p`PpH ` 0޽h ? 3333___PPT10i.v g^N+D=' | = @B +  d(  d d S   <$D 0    d S ,Z ?) <$D 0  "p`Pp d C AC:\Program Files\Common Files\Microsoft Shared\Clipart\cagcat50\PE01616_.wmfP K,$D 0H d 0޽h ? 3333___PPT10i.v`ˇ+D=' | = @B +  h(  h h S 3  <$D 0    h S H9  ) <$D 0  "p`Pp h C AC:\Program Files\Common Files\Microsoft Shared\Clipart\cagcat50\BD00028_.WMF@ 1 `,$D 0H h 0޽h ? 3333___PPT10i.v+D=' | = @B +  l(  lr l S B     ~ l s *lC  L   ` l c $A ??t H l 0޽h ? 3333___PPT10i.vM+D=' | = @B +_  vn t(  tr t S J     r t S |K     t C AC:\Program Files\Common Files\Microsoft Shared\Clipart\cagcat50\SY00451_.wmf H t 0޽h ? 3333___PPT10i.v =_+D=' | = @B +     0<- (  <r < S ]      < c $$c Y  "p`Pp < 0h n /  OA 4-total-coloring of 5-cycle 2^ < 68c R2  < s * R2  < s * X R2  < s * p@R2  < s * X0R2  < s * 0  < 0RH x/  31 2 < 0n o  31 2 < 0X ` ?  31 2 < 0S( 0  32 2 < 0r   32 2 < 0t 0p 32 2 < 0w0 X  33 2 < 0d{( X  33 2 < 0p h 33 2 < 0p  34 2H < 0޽h ? 3333___PPT10i.v@4.+D=' | = @B +F  @@@tׄp@(  r  S  G      S   "<$D 0   e 0F ,$D 0 i7Bipartite difference : 1 (Use a color for these edges.)8 28&z ]pc  g p]c ,$D 0t2 h 6"`]pt2 i 6"`pjt2 j 6"` p t2 k 6"`V p t2 l 6"`pYTB m c $D+] TB n c $DO ] TB o c $D s T TB p c $D  9 TB q c $D 9 t2 r 6"`g8  t2 s 6"`8 t t2 t 6"` 8  t2 u 6"`` 8 t2 v 6"` 8 c fz ]}` w }]`,$D 0 ]CY x# ]pY,$D 0t2 y 6"`]Ct2 z 6"`Cjt2 { 6"` C t2 | 6"`V C t2 } 6"`CY  gc  ~# ^}Z,$D 0n2  0"`g  n2  0"` t n2  0"`  n2  0"`` n2  0"`  c ZB  s *DԔj ZB  s *DԔ  ZB  s *DԔ `   3 B CDEFԔP  @  "`6   3 BCDEFԔ9qs@  "``   3 BQCDEFԔb1{^WNQ@  "` S 6   3 BCDEFԔb1zN" N@  "`jL ` ZB  s *DԔ    3 B#CDEFԔNS1#b@  "`S   3 BCDEFԔPu+e@  "`j  T ]CY # ]qYt2  6"`]Ct2  6"`Cjt2  6"` C t2  6"`V C t2  6"`CYt2  6"`^pt2  6"`pkt2  6"` p t2  6"`W p t2  6"`pZZB  s *DԔa ZB  s *DԔ ZB  s *DԔ W   C B CDEFԔP  @  "`}-p  C BCDEFԔ9qs@  "`W p  C BQCDEFԔb1{^WNQ@  "` -  C BCDEFԔb1zN" N@  "`aW ZB  s *DԔ  C B#CDEFԔNS1#b@  "`   C BCDEFԔPu+e@  "`aZ2  s *`z g9 h  9 gh,$D 0  9 5  # g9 c,$D 0n2  0"`9  n2  0"` F n2  0"` n2  0"`2 n2  0"` 5 ZB  s *DԔ< ZB  s *DԔ  ZB  s *DԔ 2   C B CDEFԔP  @  "`g    C BCDEFԔ9qs@  "` 2   C BQCDEFԔb1{^WNQ@  "` j   C BCDEFԔb1zN" N@  "`< 2 j ZB  s *DԔ    C B#CDEFԔNS1#b@  "`g j   C BCDEFԔPu+e@  "`<  t2  6"`g9  t2  6"`9 t t2  6"` 9  t2  6"`` 9 t2  6"` 9 c ZB  s *DԔjf f ZB  s *DԔ f f ZB  s *DԔ f ` f   C B CDEFԔP  @  "` 6  C BCDEFԔ9qs@  "` ` v  C BQCDEFԔb1{^WNQ@  "` f 6   C BCDEFԔb1zN" N@  "`jf ` ZB  s *DԔf f  „ C B#CDEFԔNS1#b@  "`f  Ä C BCDEFԔPu+e@  "`jf }Z2 Ą s *@ h lz -9  ń -9 ,$D 0ZB Ƅ s *DԔ9 ZB DŽ s *DԔ==9 ZB Ȅ s *DԔ  9 ZB Ʉ s *DԔ  9 ZB ʄ s *DԔ--9 |l `Xp фh`X,$D 0`2 ˄ 0```2 ̄ 0Ph`2 ̈́ 0  p`2 ΄ 0`  p`2 τ 0Xhlz `Xp ҄ ( ph ,$D 0Z2 ӄ s *``Z2 Ԅ s *PhZ2 Մ s *  pZ2 ք s *`  pZ2 ׄ s *XhH  0޽h ? 3333___PPT10.v׀+UXD|' = @B D7' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*gD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*g+`/  8)0)PM ((  r  S 0     r  S   P   . 0 F  PBipartite difference : 1 and 2 2&8 ]}c }]cz2 0 <"`]qz2 1 <"`qjz2 2 <"` q z2 3 <"`V q z2 4 <"`qYz2 5 <"`^pz2 6 <"`pkz2 7 <"` p z2 8 <"`W p z2 9 <"`pZ`B : 0DԔa `B ; 0DԔ `B < 0DԔ W  = C B CDEFԔP  @  "`}-p > C BCDEFԔ9qs@  "`W p ? C BQCDEFԔb1{^WNQ@  "` - @ C BCDEFԔb1zN" N@  "`aW `B A 0DԔ B C B#CDEFԔNS1#b@  "`  C C BCDEFԔPu+e@  "`az2 D <"`g9  z2 E <"`9 t z2 F <"` 9  z2 G <"`` 9 z2 H <"` 9 c `B I 0DԔjf f `B J 0DԔ f f `B K 0DԔ f ` f  L C B CDEFԔP  @  "` 6 M C BCDEFԔ9qs@  "` ` v N C BQCDEFԔb1{^WNQ@  "` f 6  O C BCDEFԔb1zN" N@  "`jf ` `B P 0DԔf f  Q C B#CDEFԔNS1#b@  "`f  R C BCDEFԔPu+e@  "`jf }`B S 0DԔ9 `B T 0DԔ==9 `B U 0DԔ  9 `B V 0DԔ  9 `B W 0DԔ--9 T ]pc  X# ]pc t2 Y 6"`]pt2 Z 6"`pjt2 [ 6"` p t2 \ 6"`V p t2 ] 6"`pYTB ^ c $D+] TB _ c $DO ] TB ` c $D s T TB a c $D  9 TB b c $D 9 t2 c 6"`g8  t2 d 6"`8 t t2 e 6"` 8  t2 f 6"`` 8 t2 g 6"` 8 c . ]pc   ]pc ,$D  0t2  6"`]pt2  6"`pjt2  6"` p t2  6"`V p t2  6"`pYt2  6"`g8  t2  6"`8 t t2  6"` 8  t2  6"`` 8 t2  6"` 8 c TB  c $D ] TB  c $DCf Z TB  c $D T TB B c $Df 9 TB B c $DI9 H  0޽h ? 3333___PPT10.w+zD|' = @B D7' = @BA?%,( < +O%,( < +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<*+5  //`]]\/(  x  c $     x  c $4  P    0TF  T"Bipartite difference : 1, 2, and 3# 2#h-8 ]}p ]}]pt2  6"`]qt2  6"`qjt2  6"` q t2  6"`V q t2   6"`qYt2   6"`^pt2   6"`pkt2   6"` p t2   6"`W p t2  6"`pZZB  s *DԔa ZB  s *DԔ ZB  s *DԔ W   C B CDEFԔP  @  "`}-p  C BCDEFԔ9qs@  "`W p  C BQCDEFԔb1{^WNQ@  "` -  C BCDEFԔb1zN" N@  "`aW ZB  s *DԔ  C B#CDEFԔNS1#b@  "`   C BCDEFԔPu+e@  "`at2  6"`g9  t2  6"`9 t t2  6"` 9  t2  6"`` 9 t2  6"` 9 c ZB  s *DԔjf f ZB  s *DԔ f f ZB   s *DԔ f ` f  ! C B CDEFԔP  @  "` 6 " C BCDEFԔ9qs@  "` ` v # C BQCDEFԔb1{^WNQ@  "` f 6  $ C BCDEFԔb1zN" N@  "`jf ` ZB % s *DԔf f  & C B#CDEFԔNS1#b@  "`f  ' C BCDEFԔPu+e@  "`jf }ZB ( s *DԔ9 ZB ) s *DԔ==9 ZB * s *DԔ  9 ZB + s *DԔ  9 ZB , s *DԔ--9 T ]pc  -# ]pc t2 . 6"`]pt2 / 6"`pjt2 0 6"` p t2 1 6"`V p t2 2 6"`pYTB 3 c $D+] TB 4 c $DO ] TB 5 c $D s T TB 6 c $D  9 TB 7 c $D 9 t2 8 6"`g8  t2 9 6"`8 t t2 : 6"` 8  t2 ; 6"`` 8 t2 < 6"` 8 c . ]pc  = ]pc ,$D  0t2 > 6"`]pt2 ? 6"`pjt2 @ 6"` p t2 A 6"`V p t2 B 6"`pYt2 C 6"`g8  t2 D 6"`8 t t2 E 6"` 8  t2 F 6"`` 8 t2 G 6"` 8 c TB H c $D ] TB I c $DCf Z TB J c $D T TB KB c $Df 9 TB LB c $DI9 4 ]pc  M# jhp ,$D  0t2 N 6"`]pt2 O 6"`pjt2 P 6"` p t2 Q 6"`V p t2 R 6"`pYt2 S 6"`g8  t2 T 6"`8 t t2 U 6"` 8  t2 V 6"`` 8 t2 W 6"` 8 c TB X c $D` K TB Y c $D= L TB Z c $D ? TB [ c $DIl 9 TB \ c $D 9 H  0޽h ? 3333___PPT10.w+zD|' = @B D7' = @BA?%,( < +O%,( < +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<*+,=  76pmm6(  x  c $     x  c $  P    0F  V$Bipartite difference : 1, 2, 3 and 4% 2%48 ]}p m}]pt2  6"`]qt2  6"`qjt2  6"` q t2  6"`V q t2   6"`qYt2   6"`^pt2   6"`pkt2   6"` p t2   6"`W p t2  6"`pZZB  s *DԔa ZB  s *DԔ ZB  s *DԔ W   C B CDEFԔP  @  "`}-p  C BCDEFԔ9qs@  "`W p  C BQCDEFԔb1{^WNQ@  "` -  C BCDEFԔb1zN" N@  "`aW ZB  s *DԔ  C B#CDEFԔNS1#b@  "`   C BCDEFԔPu+e@  "`at2  6"`g9  t2  6"`9 t t2  6"` 9  t2  6"`` 9 t2  6"` 9 c ZB  s *DԔjf f ZB  s *DԔ f f ZB   s *DԔ f ` f  ! C B CDEFԔP  @  "` 6 " C BCDEFԔ9qs@  "` ` v # C BQCDEFԔb1{^WNQ@  "` f 6  $ C BCDEFԔb1zN" N@  "`jf ` ZB % s *DԔf f  & C B#CDEFԔNS1#b@  "`f  ' C BCDEFԔPu+e@  "`jf }ZB ( s *DԔ9 ZB ) s *DԔ==9 ZB * s *DԔ  9 ZB + s *DԔ  9 ZB , s *DԔ--9 T ]pc  -# ]pc t2 . 6"`]pt2 / 6"`pjt2 0 6"` p t2 1 6"`V p t2 2 6"`pYTB 3 c $D+] TB 4 c $DO ] TB 5 c $D s T TB 6 c $D  9 TB 7 c $D 9 t2 8 6"`g8  t2 9 6"`8 t t2 : 6"` 8  t2 ; 6"`` 8 t2 < 6"` 8 c . ]pc  = ]pc ,$D  0t2 > 6"`]pt2 ? 6"`pjt2 @ 6"` p t2 A 6"`V p t2 B 6"`pYt2 C 6"`g8  t2 D 6"`8 t t2 E 6"` 8  t2 F 6"`` 8 t2 G 6"` 8 c TB H c $D ] TB I c $DCf Z TB J c $D T TB KB c $Df 9 TB LB c $DI9 4 ]pc  M# jhp ,$D  0t2 N 6"`]pt2 O 6"`pjt2 P 6"` p t2 Q 6"`V p t2 R 6"`pYt2 S 6"`g8  t2 T 6"`8 t t2 U 6"` 8  t2 V 6"`` 8 t2 W 6"` 8 c TB X c $D` K TB Y c $D= L TB Z c $D ? TB [ c $DIl 9 TB \ c $D 9 . ]pc  ] ]hc ,$D  0t2 ^ 6"`]pt2 _ 6"`pjt2 ` 6"` p t2 a 6"`V p t2 b 6"`pYt2 c 6"`g8  t2 d 6"`8 t t2 e 6"` 8  t2 f 6"`` 8 t2 g 6"` 8 c TB h c $D f TB iB c $D59 TB jB c $D= 9 TB kB c $D n 9 TB lB c $D 9 H  0޽h ? 3333___PPT10.w+zD|' = @B D7' = @BA?%,( < +O%,( < +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<*+'  4&,&IK%(  r  S L1       <2     b!8 ]m G]mz2  <"`]qz2  <"`qjz2  <"` q z2   <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2  <"`W p z2  <"`pZ  C BCDEFԔ9qs@  "`W p`B  0DԔz2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c   C B CDEFԔP  @  "` 6  C B#CDEFԔNS1#b@  "`f `B  0DԔ9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *D+] z2   <"`g8  z2 ! <"`8 t z2 " <"` 8  z2 # <"`` 8 z2 $ <"` 8 c z2 % <"`]pz2 & <"`pjz2 ' <"` p z2 ( <"`V p z2 ) <"`pYz2 * <"`g8  z2 + <"`8 t z2 , <"` 8  z2 - <"`` 8 z2 . <"` 8 c ZB /B s *DI9 z2 0 <"`]pz2 1 <"`pjz2 2 <"` p z2 3 <"`V p z2 4 <"`pYz2 5 <"`g8  z2 6 <"`8 t z2 7 <"` 8  z2 8 <"`` 8 z2 9 <"` 8 c ZB : s *D` K z2 ; <"`gpz2 < <"`ptz2 = <"` p z2 > <"`` p z2 ? <"` pcz2 @ <"`q8  z2 A <"`$8 ~ z2 B <"` 8  z2 C <"`j 8 z2 D <"`8 m ZB EB s *DG 9  F 0I Fs  8Tree 1 2 H 0PM   31 2 I 0P h  32 2 J 0T @  33 2 K 0Wp X x W  34 2H  0޽h ? 3333___PPT10i.LwM+D=' | = @B +$  ##EG"(  r  S ^     r  S _     8 ]m  G]m z2  <"`]qz2  <"`qjz2  <"` q z2  <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2   <"`W p z2  <"`pZ`B  0DԔa   C BCDEFԔPu+e@  "`az2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c   C BCDEFԔb1zN" N@  "`jf ` `B  0DԔf f `B  0DԔ==9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *DO ] z2  <"`g8  z2   <"`8 t z2 ! <"` 8  z2 " <"`` 8 z2 # <"` 8 c z2 $ <"`]pz2 % <"`pjz2 & <"` p z2 ' <"`V p z2 ( <"`pYz2 ) <"`g8  z2 * <"`8 t z2 + <"` 8  z2 , <"`` 8 z2 - <"` 8 c ZB . s *D ] z2 / <"`]pz2 0 <"`pjz2 1 <"` p z2 2 <"`V p z2 3 <"`pYz2 4 <"`g8  z2 5 <"`8 t z2 6 <"` 8  z2 7 <"`` 8 z2 8 <"` 8 c ZB 9 s *D= L z2 : <"`gpz2 ; <"`ptz2 < <"` p z2 = <"`` p z2 > <"` pcz2 ? <"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB DB s *D x 9  F 0p Fs  8Tree 2 2H  0޽h ? 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<"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB D s *Df  E 0( Fs  8Tree 4 2H  0޽h ? 3333___PPT10i.Ow!+D=' | = @B +$  ##EF.#(  r  S $     r  S   "  b!8 ]}m} F}]m}z2  <"`]qz2  <"`qjz2  <"` q z2  <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2   <"`W p z2  <"`pZ  C B CDEFԔP  @  "`}-p  C BQCDEFԔb1{^WNQ@  "` -z2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c `B  0DԔ f f   C BCDEFԔPu+e@  "`jf }`B  0DԔ--9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *D 9 z2  <"`g8  z2   <"`8 t z2 ! <"` 8  z2 " <"`` 8 z2 # <"` 8 c z2 $ <"`]pz2 % <"`pjz2 & <"` p z2 ' <"`V p z2 ( <"`pYz2 ) <"`g8  z2 * <"`8 t z2 + <"` 8  z2 , <"`` 8 z2 - <"` 8 c ZB .B s *Df 9 z2 / <"`]pz2 0 <"`pjz2 1 <"` p z2 2 <"`V p z2 3 <"`pYz2 4 <"`g8  z2 5 <"`8 t z2 6 <"` 8  z2 7 <"`` 8 z2 8 <"` 8 c ZB 9 s *D 9 z2 : <"`gpz2 ; <"`ptz2 < <"` p z2 = <"`` p z2 > <"` pcz2 ? <"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB DB s *D?9  E 0t Fs  8Tree 5 2H  0޽h ? 3333___PPT10i.Pw0+D=' | = @B +t   ~  (  r  S      r  S T  P    02  SThese trees are isomorphic to : 2 l      ,$D 0t2  6"`Dt2  6"`C s t2  6"`  ` s t2  6"`  " s t2   6"` s t2   6"` s t2   6"`  t2   6"` " t2   6"` ` t2  6"`C  TB  c $Dpp TB B c $D3 p TB B c $D p TB  c $Dp TB  c $Dp TB  c $D3 s 3 TB  c $D3 s p TB  c $D s TB B c $D s   0\pX< ?Done! 2,H  0޽h ? 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H. Lo      . 33.92 !Department of Applied Mathematics      . 33.2 NCTU,  . 33.2 gHsin Chu . 33.2 , Taiwan 30050  .--"System 0-&TNPP &՜.+,0    ^ pùjpmychatjp "Arial sөTahoma WingdingsSymbolBlendsMicrosoft {s边 3.0KOn the Existence of Multicolored Parallelisms of Isomorphic Spanning TreesPreliminariesFacts on Edge-ColoringsRainbow SubgraphRainbow 1-factorMulticolored SubgraphOrthogonal Latin Squares Room SquaresMulticolored Tree ParallelismK6 admits an MTPThree Conjectures on MTP!Brualdi-Hollongsworth Conjecture!Constantines Strong ConjectureTheorem (Constanine, 2002)%Theorem (Akbari, Alipour, Fu and Lo)Proof of the Main ResultAn example: m = 5An example: m = 5An example: m = 5An example: m = 5An example: m = 5An example: m = 5An example: m = 5An example: m = 5An example: m = 5An example: m = 5Concluding Remark  ϥΦr ²]pdO OLE A{ vD_㢳hlfuhlfu !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy{|}~Root EntrydO)Pictures>Current UserSummaryInformation(z6PowerPoint Document( ƳDocumentSummaryInformation8