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Theorem (Woolbright and Fu, 1998) In any (2m-1)-edge-colored K2m where m > 2, there exists a rainbow 1-factor. Open problem Can you find two edge-disjoint rainbow 1-factors in any (2m-1)-edge-colored K2m where m > 2? Bxgx gg gggggooo7gggggg% ggggoogg&g gggg*ggggoogg g@/  D    D, ,  ~:0e0e?Rectangle 3,3 S  : ZRainbow 1-factora  H , 0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' = @B +"! 0 Y Q p0 (  0 0 Jd1:?bq_Gr}_HrbMn@S 5^ `  j*"   x 0  xt<:0e0e?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!H(drs/downrev.xmlDJ@Ea71RnKZEh$ k.>ps9`:q$[ ' k*2)rq~6Bo *hB )}ՐA?=qd xZj(7LVl9.4Ӻsen5uio6G6./ D!?1Wc4q|wޢ1PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!H(drs/downrev.xmlPKvH$D  0 : Given an (n 1)-edge-colored Kn, a rainbow Hamiltonian path exists for n > n0. (!?) (I conjecture that this is a true.) How about rainbow spanning trees? We shall use multicolored spanning trees in the following slides.x gggggo gg gggog gggg g(g&   4 0  ~Q:0e0e?Rectangle 3,,S  : bRainbow Hamiltonian Patha  H 0 0޽h ? C7"IDqIi+z2O80___PPT10. ]+&§D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0x%(D' =-s6Bwipe(down)*<3<*0xD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(down)*<3<*0+" ! 0 4^(  4 4 Jc:?bq_Gr}_HrbMn@S 5^ `  j*"    4  xo:0e0e?Rectangle 2MT, : ,zBrualdi-Hollingsworth s Conjecture (1998, JCT(B)) If m > 2, then in any proper (2m 1)-edge-coloring of K2m, all edges can be partitioned into m multicolored spanning trees. (They found two.) Theorem (Krussel, Marshall and Verall, 2000, Ars Combin.) Three edge-disjoint multicolored spanning trees always exist.>x"ggegggggggoo$gggggg1ggg<g L   u        G ) 4  ~:0e0e?Rectangle 3'A   : W Known Resultsa  H 4 0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' = @B +"%! 0 !}!8!(  8 8 J:?bq_Gr}_HrbMn@S 5^ `  j*"   > 8  ~:0e0e?Rectangle 4Z  : lMulticolored Tree Parallelism e   8 jl:1?Text Box 59=M  Introduction$ 2 g     8 j`=u1?Text Box 6d  k T1 T2 T3 Color 1 x3x5 x4x6 x1x2 Color 2 x2x4 x1x5 x3x6 Color 3 x2x5 x3x4 x1x6 Color 4 x2x6 x1x3 x4x5 Color 5 x1x4 x2x3 x5x6h 2eemeemeem eememeemememem eememeememeemem eememeememeemem eememeememeemem eememeememeemem   iw ` " 8C *Group 31#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK6w,$D 02 8 V:?Oval 8 ;h  T    2 8 X+B#style.visibility<*8%(+"! 0 <m(  < < Ju?bq_Gr}_HrbMn@S 5^ `  j*"    <  ~u0e0e?Rectangle 2,S  u Ba   <   2, K2m can be (2m 1)-edge-colored in such a way that the edges can be partitioned into m isomorphic multicolored spanning trees. Constantine s Strong Conjecture (2002) If m > 2, then in any proper (2m 1)-edge-coloring of K2m, all edges can be partitioned into m isomorphic multicolored spanning trees. axggg% ggggoo ggEggg ggggg%gg gg gg  ggoo$ggg ggJC      d  + @  ~|u0e0e?Rectangle 3, S  u Y Conjectures e  H @ 0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' = @B +"! 0   D (  D D Jd??bq_Gr}_HrbMn@S 5^ `  j*"   ,  D  xD?0e0e?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Edrs/downrev.xmlDQO0I5 ZFB\6\ۥ0xrNot.|mF6c>p#y7T٫J!Ƨ( M9EEжdbwNcѕ\9Fnx"XPaK÷Nr+B#style.visibility<*D%(+"! 0  H(  H H JT??bq_Gr}_HrbMn@S 5^ `  j*"    H  ~f?0e0e?Rectangle 2,S  ? ,Multicolored Hamiltonian Cycle Parallelism -g$ c$g$ g$g$g$c$g$ c$ - j H  ~xk?0e0e?Rectangle 4 ?  K2m admits a multicolored Hamiltonian cycle parallelism (MHCP) if there exists a proper (2m+1)-edge-coloring of K2m+1 for which all edges can be partitioned into m multicolored Hamiltonian cycles. xggoo g*ggggggggooo-gg#g  H H 0޽h ? C7"IDqIi+z2O___PPT10i. 譭+D=' = @B +";! 0 B:L(  L L J8??bq_Gr}_HrbMn@S 5^ `  j*"   Y L  xL?0e0e?Rectangle 2,YS  ? #Existence of multicolored subgraphs$$g$(    < L  x?0e0e?Rectangle 3/Y ? pFirst, we consider the existence of multicolored spanning trees and the first one is easy to find. (See it?) In order to find more edge-disjoint multicolored spanning trees, we apply the following idea: Edge-switching.fce3 e\e3ee3  H L 0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' = @B +"! 0 PY(  P P JL??bq_Gr}_HrbMn@S 6^ `  j*"    P  ~p?0e0e?Rectangle 2,YS  ? MMSTa  2 P  t̼?1?Oval 5P T    2 P  t?1?Oval 6% 7  T    2 P  t?1?Oval 7 )  T    2 P  t`?1?Oval 8 t)  T    2 P  t?1?Oval 9% W  T    2  P  v?1?Oval 10% <  T    2  P  v@1?Oval 11 I T    2  P  v ?1?Oval 12% I  T    2  P  v@1?Oval 13% ]  T    B  P@  p1?Line 14}d% B P@  p1?Line 15S d B P  p1?Line 16)  B P  p1?Line 17S d B P@  p1?Line 18}% B P  p1?Line 19}w% B P  p1?Line 20j3 B P  p1?Line 21} S  P l @1?Text Box 22]:w} cx 2a    P l&@1?Text Box 23g e1< 2aia    P l*@1?Text Box 24}k  e2< 2aia    P lH5@1?Text Box 25S ,s  y1< 2aia    P lA@1?Text Box 26S  @ s  y2< 2aia    P l,W@1?Text Box 27 W  z1< 2aia    P lpX@1?Text Box 28S  s  z2< 2aia     P  vTd@0e0e?eW[HrbMn@S 29  @ @"  H P 0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B +"  ! 0   TR (  T T J`u@?bq_Gr}_HrbMn@S 6^ `  j*"    T  ~Xs@0e0e?Rectangle 3,YS  @ MMSTa    T  x@0e0e?Rectangle 2 @ * 4  is a (2m 1)-edge-coloring of K2m, and T is a multicolored spanning tree with root x. If x is incident to two leaves e1 = xy1 and e2 = xy2, then let T[x;y1,y2;z1,z2] = T  e1  e2 + y1z1 + y2z2, for some z1, z2. (Obtained by switching e1 and e2 with y1z1 and y2z2 respectively.)eeeeeeeeeemmee+eeeeeemeemeemeem eeeeeemeemeemeemeeeemeemeememeemem eemeemeemeemeememeemem eee  H T 0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B +" 9! 0 11 &'X1(  X X Jx@?bq_Gr}_HrbMn@S 6^ `  j*"    X  ~d^:0e0e?Rectangle 2,YS  @ MMSTa  2 X  th@1?Oval 5P T    2 X  tD1?Oval 6% 7  T    2 X  t<@1?Oval 7 )  T    2 X  tD1?Oval 8 t)  T    2 X  tlD1?Oval 9% W  T    2  X  v%D1?Oval 10% <  T    2  X  v0/D1?Oval 11 I T    2  X  vX9D1?Oval 12% I  T    2  X  vCD1?Oval 13% ]  T    B  X@  p1?Line 14}d% B X@  p1?Line 15S d B X  p1?Line 16)  B X  p1?Line 17S d B X@  p1?Line 18}% B X  p1?Line 20j3  X lMD1?Text Box 22]:w} cx 2a    X lWD1?Text Box 23g e1< 2aia    X l@[D1?Text Box 24}k  e2< 2aia    X lfD1?Text Box 25S ,s  y1< 2aia    X lqD1?Text Box 26S  @ s  y2< 2aia    X l}D1?Text Box 27 W  z1< 2aia    X lćD1?Text Box 28S  s  z2< 2aia   C }  XC *Group 36#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK} B #X  p1?Line 19}w% B $X  p1?Line 21} S B %XB  p1?Line 29S w B &X  p1?Line 30jS S  X lD1?Text Box 31"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Mdrs/downrev.xmlDN0EH5H 6ԭhQxD&MzF,`9:suY1PL]|L Cu72z6xpK~G>7%>͹/LuK$䙝Bt4,wVΔ*B=m[?_N=v3}V.Vfַ7#LcMpB/@^/3;L\ W?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Mdrs/downrev.xmlPK  ,$ 0 Fact: If (e1) = (y2z2) and (e2) = (y1z1), then T[x;y1,y2;z1,z2] is multicolored.W 2eeeemeememeemeemem eeeeeemeemeemeeme W  v  5  XC *Group 37#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK} ,$D 0B X  p8c?Line 32_B  X  p8c?Line 33B !XB  p8c?Line 34 5 B "X  p8c?Line 35R X l+B#style.visibility<*X%(D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*X%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(+8+0+X0 +" ! 0 0\X(  \ \ JD?bq_Gr}_HrbMn@S 6^ `  j*"    \  ~D0e0e?Rectangle 2,YS  D MMSTa   \  xxE0e0e?Rectangle 3V  D 0 Tj(i) : the j  th spanning tree which was constructed at round i. 0Oeeemmmmee4eeeegc@     : H \ 0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B +" $/! 0 ''@$%`'(  ` ` JE?bq_Gr}_HrbMn@S 6^ `  j*"   P `  ~,E0e0e?Rectangle 2,YS  E ~4Idea of finding more MST sa   `  xE0e0e?Rectangle 3 E R@Round 1. Pick any one vertex x1. Let T1(1) = x1-star.Axeemeemmeeme A 2 `  t@*E1?Oval 7 ks  T    2 `  t<1E1?Oval 8M  T    2 `  t;E1?Oval 9pP  T    2 `  v|EE1?Oval 10 @ 6  T    2  `  vOE1?Oval 11S C  T    2  `  vYE1?Oval 12 ] 9  T    2  `  v$dE1?Oval 13  a F  T    2  `  vpoE1?Oval 146 {  T    2  `  vxyE1?Oval 15s z  T    2 `  vE1?Oval 16 3 9  T    2 `  v@E1?Oval 17 =  T    2 `  vhE1?Oval 19 &  T    2 `  vE1?Oval 20)  T    2 `  vE1?Oval 21) q  T     ` llE1?Text Box 224 q %  x0< 2aia   8 ` l@E1?Text Box 23"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ӊdrs/downrev.xmlDJ@Fa;IXnFI>&v4]Ym&kH!)gr׽kDdtwB6ۛڟݎ="I\,QAI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK ,$D 0B ` j1?Line 24 % m B ` j1?Line 25 @ B ` j1?Line 27 m  B ` j1?Line 28M  m c B ` j1?Line 29= m F B `B j1?Line 30  m B `B j1?Line 31  m B ` j1?Line 32m  V B ` j1?Line 33m  3  B  ` j1?Line 34m ~  B !` j1?Line 35m  B "` j1?Line 36m   B #` j1?Line 37m  CV B $` j1?Line 38m   2 `  vE1?Oval 18 S  T    H %` 0޽h ? C7"IDqIi+z2O___PPT10.`;Lp++D4' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =%(D' =%(D-' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =-i6B wedge*<3<*`+8+0+`0 +" =y! 0 ]]P35d](  d d JpE?bq_Gr}_HrbMn@S 6^ `  j*"    d  ~E0e0e?Rectangle 2,YS  E Ba   d  xTE0e0e?Rectangle 3w  E L>Round 2. Pick x2, u. Reserve x2 as the root of the second MST.?xeemeee eemee ? 2 d XF?Oval 370 d  T    2 d X F?Oval 41 t  T    #C    dC *Group 97#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK  2 3d X@F?Oval 40   T    2 4d XF?Oval 42   T    2  d X|'F?Oval 43 d  T    2  d XT1F?Oval 44"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!8Tdrs/downrev.xmlDN0 @HCe$n,ʲ F4nI$˺GYzh,Y ד2S6ϗd!U;KN`I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!}+Tdrs/downrev.xmlDj1F }0D[bk&tlt]p>|ۆuNp +2pI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK [ ,$D 0 1d HI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!R&drs/downrev.xmlDOk@&B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.U9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!R&drs/downrev.xmlPK` d ,$D  0|B -dB L1?Line 51`  |B .dB L1?Line 52t`   /d HF?Text Box 60   i1$ 2g    0d HF?Text Box 61t d  i2$ 2g   2 d XF?Oval 65;| T    2 d XhF?Oval 67 k| T    2 d XF?Oval 68[| T    2 d XI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!=ك(drs/downrev.xmlDAk@&-)% VzB g̾ 5+zffM ^D K?_>@8L [ϞVj;7 _a.ҕ5t b{>Ⱦ1M+_]l8,Ѧz-TĤ|ʗ I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! _drs/downrev.xmlDA0 okZeEQDT<*GlKib[Y87|ٙR4T²x N.8Sp9o GXZ&/r\h/5'a*ҥ9tC[nk>:6M)GQ4  9V)}Fv57q_nR~ &w%xPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! _drs/downrev.xmlPK ,$@   0B )dB  p1?Line 86  *d H@F?Text Box 89" i2$ 2g   Xx  F   dC *Group 98#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Gdrs/downrev.xmlDOk@&J[B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.E9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Gdrs/downrev.xmlPK # ,$D 02 'd XH?Oval 95 F i  T    2 (d XH?Oval 96)F   T    2 d Z|!H?Oval 103"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!1fdrs/downrev.xmlDN@&fH0,HX6tgwAoVѴb  fqiuÕ>2)f.h؅JD !tɠڎ8uC<]%KVΓd) 6j蹦덂⼯,~lMǖl\d Nףkt>Scn ?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!1fdrs/downrev.xmlPK E ,$@ 0 T    v tY dC ,Group 109#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!,drs/downrev.xmlDOj@ IZ*.BK暄d̘w.IGӈ:WYV/"Ĺ 56IDe?Bv *(o)]^A-q3 ;Bi{-CC-K(88>CՏt|ܲz5O5\ 7@nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!,drs/downrev.xmlPKYt,$@  0 %d HP,H?Text Box 92t u1R 2gog   2 &d Z "H?Oval 106Y T    9z C   dC ,Group 116#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!-drs/downrev.xmlDOk@&JKB-=BUo3 f߆6}Pq0hSj ERtxz<2)~7{b_})] *DJWTd-lK R7\Fы4XsX~6 U[6]Oϟ<&ciV+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!-drs/downrev.xmlPKz ,,$D  0 #d HAH?Text Box 90C4___PPT10 B___PPT9$ v1'd 2gogc   2 $d ZMH?Oval 107 N T    q !d n[H1?Text Box 110"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Ldrs/downrev.xmlDJAEД$Jc:!&IPN+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!d%(Dq' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* d%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D>' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-6B#strips(downLeft)*<3<*dDC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-6B#strips(downLeft)*<3<*dD' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* d%(++0+ d0 ++0+d0 ++0+d0 ++0+!d0 +" T! 0 CwC`78hC(  h h JlH?bq_Gr}_HrbMn@S 6^ `  j*"   ' h  ~H0e0e?Rectangle 2,YS  H U Keep going! a  % h  xȎH0e0e?Rectangle 3jm H Y Round 2.  x e  2 h VH?Oval 8 t  T    2 h XH?Oval 10 d  T    2 h XH?Oval 12 $  T    2 h XH?Oval 13 d  T    2  h XH?Oval 14   T      h H@H?Text Box 17@T:  x0< 2go     h HH?Text Box 18@t :  x1R 2gog     h HH?Text Box 19 4  v1R 2gog     h H|H?Text Box 20   x2R 2gog    h HH?Text Box 21 [K  u: 2gg   2 h XI?Oval 25;| T    2 h X I?Oval 27 k| T    2 h XI?Oval 28[| T    2 h XPI?Oval 29| T    2 h X(&I?Oval 30{| T    2 h X0I?Oval 32 k | T    w  = hC *Group 66#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK  =,$@  0B 6h  p1?Line 42 = 7h H;I?Text Box 43  i1$ 2g   v   hC *Group 65#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK ,$D  0B 4hB  p1?Line 41  5h HEI?Text Box 44" i2$ 2g    h HQI?Text Box 452{ ,4___PPT10 B___PPT9$ v1'N 2go&    h H\I?Text Box 46t u1R 2gog   B h@  p1?Line 49"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Odrs/downrev.xmlDOKK1 07AiצTVfLdnA'޷jG1.(*Xǵ$d@ fӓ 6yE*p*@#ҕZ!i:mB(Zۈ[}UCqnhECz iy?UntG9ݽs~oA /s/m?cPwtvI(ȟLPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Odrs/downrev.xmlPKS  ,$@  0 x S @  hC *Group 67#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!R&drs/downrev.xmlDOk@&B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.U9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!R&drs/downrev.xmlPKS @ ,$@  0B &hB  p1?Line 48S S |B 'hB L1?Line 15`  B (hB  p1?Line 47S  B )hB  p1?Line 50ZS  B *h  p1?Line 51S z B +h  p1?Line 52S B ,h  p1?Line 53S @ IB -h  p1?Line 55S IB .hB  p1?Line 57S IB /hB  p1?Line 58 IB 0hB  p1?Line 60wS IB 1hB  p1?Line 61:S IB 2hB  p1?Line 62* IB 3hB  p1?Line 63S 2 h VtkI?Oval 9   T    2 h XuI?Oval 34Y T    2 h XdI?Oval 31  | T    2 h V̉I?Oval 60   T     h lI1?Text Box 64  y!Let T1(2)=T1(1)[x1;x2,v1;u1,v1']." 2eemmeemmeemeemeemeemeemge "  2  h VI?Oval 7   T    2 !h VI?Oval 50 d  T    w S   "hC *Group 70#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!=ك(drs/downrev.xmlDAk@&-)% VzB g̾ 5+zffM ^D K?_>@8L [ϞVj;7 _a.ҕ5t b{>Ⱦ1M+_]l8,Ѧz-TĤ|ʗ +B#style.visibility<*h%(D' =-m6Bbox(in)*<3<*hD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-m6Bbox(in)*<3<*hD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"h%(D' =-m6Bbox(in)*<3<*"hD' =%(D' =%(D/' =4@BBBB%(D' =-m6Bbox(in)*<3<*"hD' =1:Bhidden*o3>+B#style.visibility<*"h%(D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-s6Bwipe(down)*<3<*hD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-s6Bwipe(down)*<3<*h+" @! 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Bdrs/downrev.xmlDMK@aݴHiE+Eoocvc]< 3_ֈbS0 ^wQpدA$FxG .`cmq#ĥ ̡2-YL#ev"36RGtn>GWSTsn PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! 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X\[?Oval 32+ T     " Hx[?Text Box 33   zx04 2ck    # H![?Text Box 34 N  x1F 2ckc   ' $ HX$[?Text Box 35) %  xn-Z 2ckkc    % H.[?Text Box 36_ScY xnF 2ckc    & l<0[1?Text Box 38"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!)drs/downrev.xmlDN0HH\uxD`48q54hȇα| QpoA8& \\_ͱ[vI¡Bm}%e[&'NŘNH"JibOݗU`yי]a"?zUo:qz}"y)0& rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!)drs/downrev.xmlPK A ~ ,$ 0 gU" 2e    ' l|M[1?Text Box 39"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*drs/downrev.xmlDN0HHܨHVPT o:MXบ|rF CXA>@Nw(w BDh X.Xiw ؈$P6ƾ2-Y 'vtbLoxNrkdeqZhUKie}i^.|Vf|4gI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Hdrs/downrev.xmlDOk06Z&Eaݞɳl^JǏj3Fڱt \*8 |@68&a<07n/P>GUm.YSG:!®!FΒd.-*liWVt?x97Y[ԟũLzyK&_o n"PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Hdrs/downrev.xmlPKC& T $D 0 [   \A ? ?Object 46"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!bdrs/downrev.xmlDO]KA} |Y"WGP) /;ݭ{יq=lѻFuC-l`4@bk. |@l&?`1anobR T1֡aJKOc)FdٽvXsj8Y7_ֿRec1q *RgOFw[PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!bdrs/downrev.xmlPK Gc 8 $D  0H ( 0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+d@D' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* Fr%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* r%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*' %(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+ 0 ++0+& 0 ++0+' 0 +"  ! 0 &(    Jhd?bq_Gr}_HrbMn@S 6^ `  j*"      ~d0e0e?Rectangle 2,YS  d MMSTa     xn0e0e?Rectangle 3 d Theorem 3.1 Let  be an arbitrary (2m 1)-edge-coloring of K2m and x0 be an arbitrary vertex. Then, there exist mutually edge-disjoint multicolored spanning trees, each of which contains a pendent vertex x0. rt eeeggggggooggogggogc   0 lA ? ?Object 5pT (  nH  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B +" m! 0 tl(    Jn?bq_Gr}_HrbMn@S 5^ `  j*"   W   xn0e0e?Rectangle 2,YS  n #Existence of Multicolored Subgraphs$$c$(    p   x!n0e0e?Rectangle 3Y  n 2. Unicyclic Spanning Subgraph"g(3@       H  0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' = @B +"6 ! 0 = 5 (    J.n?bq_Gr}_HrbMn@S 6^ `  j*"       ~-n0e0e?Rectangle 2,YS  n NMUSSa  \   xH;n0e0e?Rectangle 3I  n bCorollary 3.2 Letbe an arbitrary (2m 1)-edge-coloring of K2m 1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS).eeeggggooo=gg gg gggggcZ  u         @  j0Un1?Text Box 4kM "Existence of Multicolored Subgraph$# 2#g(     0 lA ? ?Object 52g=(  nH  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B +" ! 0 |sts ./ s(    Jdrn?bq_Gr}_HrbMn@S 6^ `  j*"   2  3 |n1?Oval 38  T        ~Ȉn0e0e?Rectangle 2,YS  n NMUSSa  "   xn0e0e?Rectangle 3 n VProof. xe  @  j(n1?Text Box 4kM "Existence of Multicolored Subgraph$# 2#g(    2   tn1?Oval 5 wc  T    2   t,n1?Oval 6S  T    2   tn1?Oval 7 =c  T    2   t n1?Oval 8 -s  T    2   tHn1?Oval 9pk T    2   v8n1?Oval 10 g  T    2   v@n1?Oval 11 M   T    2   vhn1?Oval 126 h  T    2   vn1?Oval 13s g  T    2   vq1?Oval 14 =  T    2   vT q1?Oval 15) -  T    2   vq1?Oval 16Z&  T    2   vq1?Oval 17 {  T    2   vT*q1?Oval 18 k  T    2   v|4q1?Oval 19 M V  T    8  l0>q1?Text Box 20"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Wdrs/downrev.xmlDJ1&3`ǦE+EwҟNfHb: .ty86E"A83pt=4\(f}}ڸ3)b'2CZd1,Dy1GIvR8p~qmOjO\(wti"ǜ֦f4ZV ˇC$! fݬ rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Wdrs/downrev.xmlPK c a  ,$ 0 x0< 2aia   2   vAq1?Oval 37"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Fdrs/downrev.xmlDQK0Co.iJDكl涍6IM÷\f`-0SvꧻXhΒ3XX)w{c bCNJd0,H6uc)s! nPH=5߇@?芍]Ky{3oEe|i_^<e}>z"y I0&U_PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Fdrs/downrev.xmlPK  a  ,$@ 0 T    m  lVq1?Text Box 39"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!w drs/downrev.xmlDN0DHuQ;A %[AP;\$_MX,r43:d4)Y L#:5Nay B,IPfrgq:!6(OלǶ'q<]0r Wn4/=5=ݷ~-ބ UHy7,%uOPB</_aP[,u*/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!w drs/downrev.xmlPK PP p 2(2m 1)-edge-colored K2m-1t 2   aiii   ww % 3  C *Group 45#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK% 3 ,$D  0B +  p1?Line 42 % 3 B ,  p1?Line 43 3 F B -  p1?Line 44 3   lYq1?Text Box 46"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Vdrs/downrev.xmlDn0ET"A-J!#MZ];:Wgf9+!`dk:H8/S`!U% 7 Z>>,Tjw4Ta bCPd0L\O6u' }Õks! 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C7"IDqIi+z2O___PPT10i. ]+D=' = @B +"$ ! 0 $(  r  S -,YS   r  S xI  H  0޽h ? C7"IDqIi+z2O80___PPT10.S*vI! 0 PHP(    J5r?bq_Gr}_HrbMn@S 5^ `  j*"   -   x\7r0e0e?Rectangle 2,YS  r a Reference a   v   x`Cr0e0e?Rectangle 3/Y r [1] S. Akbari, A. Alipour, H. L. Fu and Y. H. Lo, Multicolored parallelisms of isomorphic spanning trees, SIAM J. Discrete Math. 20 (2006), No. 3, 564-567. [2] R. A6. Brualdi and S. Hollingsworth, Multicolored trees in complete graphs, J. Combin. Theory Ser. B 68 (1996), No. 2, 310-313. [3] G. M. Constantine, Multicolored parallelisms of isomorphic spanning trees, Discrete Math. Theor. Compu. Sci. 5 (2002), No. 1, 121-125. [4] G. M. Constantine, Edge-disjoint isomorphic multicolored trees and cycles in complete graphs, SIAM J. Discrete Math. 18 (2005), No. 3, 577-580. [5] H. L. Fu and Y. H. Lo, Multicolored parallelisms of Hamiltonian cycles, Discrete Math. 309 (2009), No. 14, 4871-4876."xg      A      1 H  0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' = @B +" ! 0 `S(    Jar?bq_Gr}_HrbMn@S 5^ `  j*"   2   ~dr0e0e?Rectangle 2,YS  r `Continued &  a     xd~r0e0e?Rectangle 3/Y r [6] H. L. Fu and Y. H. Lo, Multicolored isomorphic spanning trees in complete graphs, Ars Combinatoria, to appear. [7] H. L. Fu and D. E. Woolbright, On the exists of rainbows in 1-factorizations of K2n, J. Combin. Des. 6 (1998), 1-20. [8] J. Krussel, S. Marshal and H. Verral, Spanning trees orthogonal to one-factorizations of K2n, Ars Combin. 57 (2000), 77-82. [9] H. J. Ryser, Neuere Probleme der Kombinatorik, in: Vortrge ber Kombinatorik, Oberwolfach, Mathematisches Forschungsinstitute Oberwolfach, Germany, 24-29. xggooggoogV     $   <      :                                 H  0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' = @B +"! 0 pm(    Jr?bq_Gr}_HrbMn@S 5^ `  j*"   8   xܦr0e0e?Rectangle 2,V   r lThank you for your attention!c0  H  0޽h ? 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