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A good place to start is considering the complete graphs which are properly edge-colored with minimum number of colors, their chromatic indices.,& $ %  Q  J@O ?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.Y ޠ+D=' = @B +"! /'P(  Q  J 38 ?bq_Gr}_HrbMn@S 6^ `M<4___PPT9  *,(" dc +   ~b. 0e0e?Rectangle 5,YS   Y Preliminariesa     x@18 0e0e?Rectangle 7" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!MmM~=drs/shapexml.xml[n6;.8$F`Y2ekMQ*E;NbOEwe}-r(mD uϙ8ÿE,9WYȎ_}_=.dqt-^1Hy<񇓴*ˬvi\΂ Y>ID^iDո*qFG(*E?ESr>HB弯hZf$@Xp!$f95{)vd/U\L?S*p6( Kb%h%Z҉SP9LFH`"ǢkzCj-jhjvfgEqzIzը{h4 oj59@JAbRbm8wTHNCPz; SįoUu Gm"FnY|ZlB#AL L_/nJ= ds &,2mE.k$d@L=3EbYjՓk>1S=W]!9H,&@ވl8(6صYO&R*ChPm&g2@'#KA'4ڛ3 Zo8㲛EƺY]6rypv!GR0 gFN" ƍ,߀K2陋&]GTx0q_jƑb;i cmF`Ngq'#3d,}` 3IZZ Zaɘh e20!ߛrQ> K ID i M^_%Ih£Hi,3"q $eҍXbl0HD4Dd866K0,֬'FcfL6pwVԜmd|?rUXt%O[=6e)(T* ><JMY[9a ;Bk{_!Uj^ymT,GZh~1&/2-̏N$u_8E,Vܸ:}=h9naNY E7/5/4sN8^ |;|8wwŌ& F3~֩衰uB3HiZ4 7ܙjkǔVtX];'D6jKv71@~!uxW݅Ǐ^YjtasFdqܱ:אzLp4k&g{fw9ѵk~җpW6&A:ixrCzGgA6&m)6|{8/cE0KOPK!WH 7drs/downrev.xmlDN0H}kQJ!ԭTHܶ6qv@YͧYmFݲ90&TV*S ؕ{`>ZCa\0v09P(1>CM]ƹ觶#:!$ vP]Z^w~ئc,׋7#@c<}1O.cK-oS2G 17b_PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!MmM~=)drs/shapexml.xmlPK-!WH 7drs/downrev.xmlPKws H$@  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. In this talk, all colorings are proper, i. e. incident edges receive distinct colors. A proper 3-edge-coloring of K4 x!&&&& &&&&& &&& &&*&&&&)&&$$,g*G   Z 0 s A ?5%8c8c    ?Object 8#"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.xmlDOj0 aW: x84giYs%v>2x6@> qtF)AsL ~(f4Yaݍ?h8E#R 41nb8q[ z#[ \dY!-iPuZ3sʲ=,?^JM;Hc|.$r PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!,vdrs/downrev.xmlPK T $D  C7"IDqIi+z2O<4___PPT10. ]+D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(Du' = @B%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+" ! `(  '   jO 0e0e? jL 5,YS   iChromatic Indices"$     z $ 0e0e?gQ[HrbMn@S 6Y  It is well-known that the complete graph of order n, Kn, is of Class 1 if and only if n is even. That is, c (Kn) = n  1 if n is even and c (Kn) = n if n is odd. A graph G is of Class 1 if its chromatic index is equal to "(G) its maximum degree.5  (3  a Q  JO ?bq_Gr}_HrbMn@S 4^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.-Sq+D=' = @B +"M! EEp**~E(  Q  Jb ?bq_Gr}_HrbMn@S 5^ `M<4___PPT9  *,(" dc     xO 0e0e?Rectangle 2"PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!2drs/shapexml.xmlYn7;dY%6 j+mnI,!r*C.9R- FQB cş!93|:Wl% Τ$󈿹Y't"Teξڒa2 am̅}YRc.-L.f.R;pPN縝L3lWrb^M ˒Gi+C۵XX!ƸE]#na5.7g &ύ)nR$qnQUCSҥ\0wWBYq辎x; O0! `g&:5jv*E2}x<>9>:tH1kbN,Dh=4)4Vʡ[_&ZM;Pa95?WHԏnzPN褋i̚3/ 1:ƮwC,R1 -z}#LɨEbb5.d'=@(.lَf\1BGw坠V2om$Db6! 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(A multicolored graph is also known as a rainbow, hetero-chromatic or poly-chromatic graph.) x-& &&a&&&(&&gc  f   `O 0e0e?Rectangle 3,S   dMulticolored graph g$ /w : @  C *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 :@ ,$@R  bD ?AutoShape 40 j D<4___PPT9 l, " dc R  bD ?AutoShape 41: ) D<4___PPT9 l, " dc R  b D ?AutoShape 42 @ ) D<4___PPT9 l, " dc R  bD ?AutoShape 43G)D<4___PPT9 l, " dc R  bD ?AutoShape 44 a D<4___PPT9 l, " dc vB B FjJ?Line 45g = vB  FjJ?Line 46g wvB  FjJ?Line 47w0 vB  FjJ?Line 480  vB  FjJ?Line 49=  vB B FjJ?Line 50w =vB  FjJ?Line 51= 0 vB  FjJ?Line 52g 0 vB  FjJ?Line 53g  vB   FjJ?Line 54w  E ! H @ ?Text Box 55D<4___PPT9 y10 " d2g E " H`D ?Text Box 56U ) D<4___PPT9 y10 " d2g E # HD ?Text Box 57-  D<4___PPT9 y20 " d2g E $ HD ?Text Box 58  : D<4___PPT9 y20 " d2g E % HO ?Text Box 59$  D<4___PPT9 y30 " d2g E & H`b ?Text Box 60M 6  D<4___PPT9 y30 " d2g E ' Hb ?Text Box 61j D<4___PPT9 y40 " d2g E ( H"@ ?Text Box 62OH  D<4___PPT9 y40 " d2g E ) H38 ?Text Box 63 D<4___PPT9 y50 " d2g E * H: ?Text Box 64:  D<4___PPT9 y50 " d2g   l@b 1?Text Box 65"( " PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Ļddrs/shapexml.xmlXnF;,x 4%Blj`Dr,Vb\+[vW(gJѷ.)RSqhZݙffw/%'LS9 q>uޜ%n(:L9/E=Q5Sgu= Td%Uϫ xWnL1J>v%xxw|tt~ɲ2&nÏB,RY*Ӫ@<)4A Y{mVA6Tp:4N򩓘 `[ "85ɇ: b7Ec7L¡0ya'& Z~I CyAJ)OZc+,4ԉ6Bt~"2kH^EUΠ<_Lt@L p*'X0a!ˮ "b1z}()| \_ *Ji>=FgP^L'TӕH6x:=dU+=M EMMCX V((lK^Fz#Xo6|UR mi pgS .Z dR15V 7Uڤ wk.3hA8:xDOt70#)tۥL+5qzoPK!2udrs/downrev.xmlDAO1F&fHBqb4rtaۮme=N}yuov!& Im+u"Z$J֫s:!V,IlQ@csʚ k&7+.=vIn>Ͳ97mZP*MRg!n X>?w7;Wev?_<SoC$/ ܄~PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!Ļd)drs/shapexml.xmlPK-!2udrs/downrev.xmlPK    p$D<4___PPT9 4A multicolored 5-cycle in a properly edge-colored K5R5 " d23&.&5  v t t  C *Group 78#"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 @@ ,$D|B B LԔ?Line 79 w |B   LԔ?Line 80 |B   LԔ?Line 81j|B   LԔ?Line 82j M |B   LԔ?Line 83w M E   HD ?Text Box 84D<4___PPT9 y10 " d2g E  HD ?Text Box 85 t D<4___PPT9 y20 " d2g E  Hm? ?Text Box 86 = D<4___PPT9 y30 " d2g E  HD ?Text Box 87 ? 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IA )(   Q  JD ?bq_Gr}_HrbMn@S 5^ `M<4___PPT9  *,(" dc h   x D 0e0e?Rectangle 2"  PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!:Ԅ:drs/shapexml.xml[n6;.FEm#0,r h(NbOEwe}-EDiQp> ɡ,Yst/']Wi1'\yIe7"ao($2]6:T;!O^G"rTc%!5( Zr_ct%x(/]֪#yQ/ &pl#w2V=&ņ#fRǘS cܪ/cUS%:&h|˚v}pl;KSM̔ygNyk >h #qQߪ4k pAl5arbfiY;zy9Q47c4-s+y'̹ ~sl5m22jfY#Y̸tkiFf1}9]"*ї+bhٜG)!~ MZbND V#Ea0c{1~כ{0B |9vY%j0g,K>ަصm#x_iER&#hPg&^ 5#ˀ$O%;pKЊxTxcMq]ڬKХY-d2iYf4ޮx<=Lh]hfI&2aNkw@XoSKEmaL)2I##xrt=0gwYdeBV7i 5#C&H1 6p&ŜP4.`JE"IV;oO&KHCXhn*<Jc(MB}nz~e ØJ(Th2ۀ&F`P-*y}y4"S7iڬ t^*|o?ǯ;VU>ms7h*7 p,Xع6TH54ǂ >#/>5Lb>mJ 68zε@:εcJ,Yi\roT! *41=*^R]rNI1=/^H7A=bLژV" :M.K q+یK,}_]~XE4أvisx%3f2Z W,>/2Bj9sB£,pԅ0FNȦM.t[{, 2, there exists a rainbow 1-factor. This result has been further improved to complete uniform hypergraphs by Saad El-Zanati et al. later. It remains unknown for finding two or more edge-disjoint rainbow 1-factors. oxixLxx&&&%&&% &&&&..&&&&L&&&&&L"g*  f D(   ~`D 0e0e?Rectangle 3,3 S   V Motivation a   C7"IDqIi+z2O___PPT10i. ]+D=' = @B +"! $(  $ $  j!@ 0e0e? jL 1,YS   ` Follow-ups     D $  z@D 0e0e?gQ[HrbMn@S 2Y  vDavid looked at finding a long rainbow path, hopefully a Hamilton rainbow path. He believed that a fairly long rainbow path can be found. (There are results on this part.) I looked at finding more rainbow 1-factors which are edge-disjoint. I can not find even two such 1-factors after 20 years of  looking if the coloring is arbitrarily given.Y9"&r"&S"&"&"X  Q $ J` ?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.Y+D=' = @B +"! ((  (* (  j@b 0e0e? jL 1,YS   lMore Rainbow 1-factors   , (  z`48 0e0e?gQ[HrbMn@S 2]'<  ^If we are allowed to design an edge-coloring for K2m, then we are able to find (2m  1) rainbow 1-factors inside K2m. 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@13dc$K9u72x$Q-)I*<(q9H[TP7E,`Tw ,m]'}gh9]g(е4bncӈa7r‘'g_RQd K)9Ub1vW(6@kہވJdmhAY;:<{ڙyh޾&,I(nxpŢTnE>wb:> aMXA0,Kp;TNj=&S"s>!zb(AmF%]7`lySϷ±X# JƬquW7[}nF&y&0XGfN" (ג$cg 35E%vFs4~!Թ3Й X9dZrR4ӑUDZJ9k{anIƷ<)s0{Q2$2`x1Dwi+2>eTm{XT,u%Y~c_6]M . y^ҔĐW_HX*(3/$d:ޒ*ZHB{dyʳxq@!Pk^u] {"8i곕?.>'bc }0~!!*`(Ο+2.D"phd2 -{и_|x ⡡z д SF ⡡;h\؏[/hL9SKUMJ2@h^4N@]t27hVrYvAVxvxq("U$h x[rQ7Y b!#JS`kf\[td_3o,&kqu?^߿tqo S/'9J:mx /^/#G&j !diQ֨(kԑyDOZ V ;UPK!drs/downrev.xmlDN0 @HCd$.+ lBLkJJ8zֳl1V(;Q\u*OS1zG a1?=a}6tئJBXG#lcسܶ2ϲiv|`GKCe[̾6mxbRlhHejZ ?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!IJE)drs/shapexml.xmlPK-!drs/downrev.xmlPK gD<4___PPT9 "7 edge-disjoint rainbow matchings!,# " d#&"   C7"IDqIi+z2O___PPT10i.3`FU+D=' = @B +"k! rj0R(  0* 0  j5 0e0e? jL 1,YS   lMulticolored subgraphs    0  z2 0e0e?gQ[HrbMn@S 2]Y<  'For convenience in locating the references in literatures, we shall use Multicolored subgraphs in what follows. Of course, Rainbow is more beautiful!RH $ $$   Q 0 J@6 ?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  0 jA?? imagesWGr 4images.jpg V  C7"IDqIi+z2O___PPT10i.Y +D=' = @B +" !   4 (  4 4 zA?? S__5480450WGr 1S__5480450.jpgo  4 D`9 ?eW[eJX 2"( " PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!)dVdrs/shapexml.xmlXKoF x 9A AŊ\ZKbe=衧@EEgve;M`B iI.wg2g UV=umH2~4_Fж*IxBX>|믞c^ʱ=u:U<9%.-DN$܊N)hE$6Yw~''`)~2-'%vC$m//~|篗'˷;f*~eN&E|\ȧHr ebwN8STQ2rV?\PP' Y(**X}wp`pCȈ.Ä p{~϶b1p.QR2BIpf)* -, mAc4%'J-p;^Dcw5ґ.H ʸ9F"oC7'N9n#oy?ݼ`4ϒ򶙼Y,HӸ;YL;Ƈ!<+e .Uh˄uB؎e5.Vw?8As'?Hܲ? Ik$H3e;34%)dkBԄd&d!zA]aB êPK!drs/downrev.xmlDN0EH5H u+R>`qHliS X\鼳8Qw ʕ 1il#0]_M1tڥR\X{%g(xfm ze3Vד&w,T{zKz%We4N@^.[Z g?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!)dV)drs/shapexml.xmlPK-!drs/downrev.xmlPK D<4___PPT9 Taipei City with two rainbows> " d &(&(   C7"IDqIi+z2O___PPT10i.9Z<~+D=' = @B +"! 8(  8' 8  j`4 0e0e? jL 1,YS   iPrescribed coloring    8  z 8 0e0e?gQ[HrbMn@S 2"} w PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!!.drs/shapexml.xmlZOo6wx-\;οƈR4 0 Nc@˴Z4vv)0.; (ze}4ȷ"9MULt||G?]ƑJ Q('>{ym3A$NA!%U/OJ=JX`Sa IHkN IE"0宽NOUg# !Z@J(:ƊGQļngc@8Ar*m ly3;,E`omUmz*9T(H3$e ș+I s8"E215㥕V <_5"d-xï/~ᷟtq56Vx'Ѕbt5ʡeN=ٗ+ȉ@O~n&5}(>s8> \9adK???Jr b[]OZ |%X`mX~=q1ClnsV? -FiX[qm3B9i?Xf ŝ܅t>&&SJ/3]mY#WI4לtyW QKQzooZ_wFbFΑһ7ZŽ^޶B-O0k$gtxҜ g,n;[ ӓgs7Y>ܳVg*Kl`˹z`uW(\¦Ӯ-JUVD5ˬ^@}Pr=@!G}8v0q7 ,M/^n vV7PK! D̊drs/downrev.xmlDOK@GaKi7ZK춈 4 m~{z~j0ڲIRa2Ycc\jy}L۞B)} LJ_TdOlK:!a妑I2k \Qqڝ!ߟN|ϷvR7#@CW[,q/_[􁜂k#PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!!.)drs/shapexml.xmlPK-! D̊fdrs/downrev.xmlPKlY  8Constantine s Weak Conjecture (2002) For any m > 2, K2m can be (2m 1)-edge-colored in such a way that the edges can be partitioned into m isomorphic multicolored spanning trees. This conjecture has been verified later in 2006. (Work jointly with S. Akbari, A. Alipour and Y. H. Lo.) $$ $$$$,, $$E$$$ $$1$9 8C     Q 8 J0@ ?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.8J)+D=' = @B +"_! 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" d2e  C7"IDqIi+z2Or j ___PPT10J . ]+XD ' = @B Di ' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D[' = @B%(D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-@%(DN' = @B%(D' = @B%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(+p+0+@ ++0+-@ +" u! ttmmDkt(  D% D  x=0e0e?Rectangle 2,YS   YAn example: m = 5  X D  x` 0e0e?Rectangle 3b >6___PPT9 F nZ Z D F ?Text Box 4F D<4___PPT9 $Bipartite difference : 1, 2, 3 and 4,% " d2%c% Dpz ]}p DC ,Group 109"}]pL2 D \` ?Oval 5]qD<4___PPT9 l, " dc L2 D \ ?Oval 6qjD<4___PPT9 l, " dc L2 D \h?Oval 7 q D<4___PPT9 l, " dc L2  D \{(?Oval 8V q D<4___PPT9 l, " dc L2  D \h?Oval 9qYD<4___PPT9 l, " dc N2  D ^|(?Oval 10^pD<4___PPT9 l, " dc N2  D ^`h?Oval 11pkD<4___PPT9 l, " dc N2  D ^ }(?Oval 12 p D<4___PPT9 l, " dc N2 D ^h?Oval 13W p D<4___PPT9 l, " dc N2 D ^}(?Oval 14pZD<4___PPT9 l, " dc |B D LԔ?Line 15a |B D LԔ?Line 16 |B D LԔ?Line 17 W  D # Z@AB CDEFQRUVWXԔ?P  @  P  Freeform 18}-p D # Z@ABCDEFQRUVWXԔ?9qs@  Freeform 19W p D # Z@ABQCDEFQRUVWXԔ?b1{^WNQ@  b{QQFreeform 20 - D # Z@ABCDEFQRUVWXԔ?b1zN" N@  bNFreeform 21aW |B D LԔ?Line 22 D # Z@AB#CDEFQRUVWXԔ?NS1#b@  #b#Freeform 23  D # Z@ABCDEFQRUVWXԔ?Pu+e@  Freeform 24aN2 D ^h?Oval 25g9  D<4___PPT9 l, " dc N2 D ^`h?Oval 269 t D<4___PPT9 l, " dc N2 D ^ q?Oval 27 9  D<4___PPT9 l, " dc N2 D ^q?Oval 28` 9 D<4___PPT9 l, " dc N2 D ^q?Oval 29 9 c D<4___PPT9 l, " dc |B D LԔ?Line 30jf f |B  D LԔ?Line 31 f f |B !D LԔ?Line 32 f ` f  "D # Z@AB CDEFQRUVWXԔ?P  @  P  Freeform 33 6 #D # Z@ABCDEFQRUVWXԔ?9qs@  Freeform 34 ` v $D # Z@ABQCDEFQRUVWXԔ?b1{^WNQ@  b{QQFreeform 35 f 6  %D # Z@ABCDEFQRUVWXԔ?b1zN" N@  bNFreeform 36jf ` |B &D LԔ?Line 37f f  'D # Z@AB#CDEFQRUVWXԔ?NS1#b@  #b#Freeform 38f  (D # Z@ABCDEFQRUVWXԔ?Pu+e@  Freeform 39jf }|B )D LԔ?Line 409 |B *D LԔ?Line 41==9 |B +D LԔ?Line 42  9 |B ,D LԔ?Line 43  9 |B -D LԔ?Line 44--9 Z ]pc  .DC *Group 45"]pc N2 _D ^q?Oval 46]pD<4___PPT9 l, " dc N2 `D ^(?Oval 47pjD<4___PPT9 l, " dc N2 aD ^h?Oval 48 p D<4___PPT9 l, " dc N2 bD ^ ?Oval 49V p D<4___PPT9 l, " dc N2 cD ^5 ?Oval 50pYD<4___PPT9 l, " dc vB dD F?Line 51+] vB eD F?Line 52O ] vB fD F?Line 53 s T vB gD F?Line 54  9 vB hD F?Line 55 9 N2 iD ^ ?Oval 56g8  D<4___PPT9 l, " dc N2 jD ^ ?Oval 578 t D<4___PPT9 l, " dc N2 kD ^ ?Oval 58 8  D<4___PPT9 l, " dc N2 lD ^@ ?Oval 59` 8 D<4___PPT9 l, " dc N2 mD ^ ?Oval 60 8 c D<4___PPT9 l, " dc  ]pc  /DC *Group 61"]pc ,$D N2 PD ^`q?Oval 62]pD<4___PPT9 l, " dc N2 QD ^?Oval 63pjD<4___PPT9 l, " dc N2 RD ^q?Oval 64 p D<4___PPT9 l, " dc N2 SD ^@?Oval 65V p D<4___PPT9 l, " dc N2 TD ^?Oval 66pYD<4___PPT9 l, " dc N2 UD ^?Oval 67g8  D<4___PPT9 l, " dc N2 VD ^?Oval 688 t D<4___PPT9 l, " dc N2 WD ^?Oval 69 8  D<4___PPT9 l, " dc N2 XD ^ ?Oval 70` 8 D<4___PPT9 l, " dc N2 YD ^?Oval 71 8 c D<4___PPT9 l, " dc vB ZD F?Line 72 ] vB [D F?Line 73Cf Z vB \D F?Line 74 T vB ]DB F?Line 75f 9 vB ^DB F?Line 76I9  ]pc  0DC *Group 77"jhp ,$D N2 AD ^@q?Oval 78]pD<4___PPT9 l, " dc N2 BD ^q?Oval 79pjD<4___PPT9 l, " dc N2 CD ^q?Oval 80 p D<4___PPT9 l, " dc N2 DD ^ q?Oval 81V p D<4___PPT9 l, " dc N2 ED ^@{(?Oval 82pYD<4___PPT9 l, " dc N2 FD ^`>?Oval 83g8  D<4___PPT9 l, " dc N2 GD ^z(?Oval 848 t D<4___PPT9 l, " dc N2 HD ^?Oval 85 8  D<4___PPT9 l, " dc N2 ID ^ ~ ?Oval 86` 8 D<4___PPT9 l, " dc N2 JD ^?Oval 87 8 c D<4___PPT9 l, " dc vB KD F?Line 88` K vB LD F?Line 89= L vB MD F?Line 90 ? vB ND F?Line 91Il 9 vB OD F?Line 92 9  ]pc  1DC *Group 93"]hc ,$D N2 2D ^@q?Oval 94]pD<4___PPT9 l, " dc N2 3D ^q?Oval 95pjD<4___PPT9 l, " dc N2 4D ^w(?Oval 96 p D<4___PPT9 l, " dc N2 5D ^`q?Oval 97V p D<4___PPT9 l, " dc N2 6D ^`q?Oval 98pYD<4___PPT9 l, " dc N2 7D ^q?Oval 99g8  D<4___PPT9 l, " dc P2 8D `q?Oval 1008 t D<4___PPT9 l, " dc P2 9D `q?Oval 101 8  D<4___PPT9 l, " dc P2 :D `q?Oval 102` 8 D<4___PPT9 l, " dc P2 ;D `@q?Oval 103 8 c D<4___PPT9 l, " dc xB DB H?Line 106= 9 xB ?DB H?Line 107 n 9 xB @DB H?Line 108 9  C7"IDqIi+z2O___PPT10i.w+D=' = @B +"Ib! PaHa NNH0a(  H% H  x2@ 0e0e?Rectangle 2,YS   YAn example: m = 5  Z H  x@ 0e0e?Rectangle 4bb >6___PPT9 H n"Qx ]m HC *Group 71"]mL2 H \U ?Oval 6]qD<4___PPT9 l, " dc L2 H \@[ ?Oval 7qjD<4___PPT9 l, " dc L2 H \ S ?Oval 8 q D<4___PPT9 l, " dc L2 H \\ ?Oval 9V q D<4___PPT9 l, " dc N2 H ^P ?Oval 10qYD<4___PPT9 l, " dc N2 H ^W ?Oval 11^pD<4___PPT9 l, " dc N2 H ^ ?Oval 12pkD<4___PPT9 l, " dc N2 H ^] ?Oval 13 p D<4___PPT9 l, " dc N2 H ^ ?Oval 14W p D<4___PPT9 l, " dc N2 H ^`^ ?Oval 15pZD<4___PPT9 l, " dc  H # Z@ABCDEFQRUVWXԔ?9qs@  Freeform 16W p|B H LԔ?Line 17N2 H ^1?Oval 18g9  D<4___PPT9 l, " dc N2 H ^ ?Oval 199 t D<4___PPT9 l, " dc N2 H ^q?Oval 20 9  D<4___PPT9 l, " dc N2 H ^ ?Oval 21` 9 D<4___PPT9 l, " dc N2 H ^??Oval 22 9 c D<4___PPT9 l, " dc   H # Z@AB CDEFQRUVWXԔ?P  @  P  Freeform 23 6 !H # Z@AB#CDEFQRUVWXԔ?NS1#b@  #b#Freeform 24f |B "H LԔ?Line 259 N2 #H ^q?Oval 26]pD<4___PPT9 l, " dc N2 $H ^ ?Oval 27pjD<4___PPT9 l, " dc N2 %H ^q?Oval 28 p D<4___PPT9 l, " dc N2 &H ^U ?Oval 29V p D<4___PPT9 l, " dc N2 'H ^R ?Oval 30pYD<4___PPT9 l, " dc vB (H F?Line 31+] N2 )H ^@V ?Oval 32g8  D<4___PPT9 l, " dc N2 *H ^` ?Oval 338 t D<4___PPT9 l, " dc N2 +H ^ ?Oval 34 8  D<4___PPT9 l, " dc N2 ,H ^q?Oval 35` 8 D<4___PPT9 l, " dc N2 -H ^?Oval 36 8 c D<4___PPT9 l, " dc N2 .H ^`?Oval 37]pD<4___PPT9 l, " dc N2 /H ^`?Oval 38pjD<4___PPT9 l, " dc N2 0H ^?Oval 39 p D<4___PPT9 l, " dc N2 1H ^?Oval 40V p D<4___PPT9 l, " dc N2 2H ^?Oval 41pYD<4___PPT9 l, " dc N2 3H ^?Oval 42g8  D<4___PPT9 l, " dc N2 4H ^@?Oval 438 t D<4___PPT9 l, " dc N2 5H ^?Oval 44 8  D<4___PPT9 l, " dc N2 6H ^ ?Oval 45` 8 D<4___PPT9 l, " dc N2 7H ^?Oval 46 8 c D<4___PPT9 l, " dc vB 8HB F?Line 47I9 N2 9H ^?Oval 48]pD<4___PPT9 l, " dc N2 :H ^`?Oval 49pjD<4___PPT9 l, " dc N2 ;H ^S ?Oval 50 p D<4___PPT9 l, " dc N2 H ^?Oval 53g8  D<4___PPT9 l, " dc N2 ?H ^?Oval 548 t D<4___PPT9 l, " dc N2 @H ^ ] ?Oval 55 8  D<4___PPT9 l, " dc N2 AH ^?Oval 56` 8 D<4___PPT9 l, " dc N2 BH ^@?Oval 57 8 c D<4___PPT9 l, " dc vB CH F?Line 58` K N2 DH ^?Oval 59gpD<4___PPT9 l, " dc N2 EH ^_ ?Oval 60ptD<4___PPT9 l, " dc N2 FH ^?Oval 61 p D<4___PPT9 l, " dc N2 GH ^Z ?Oval 62` p D<4___PPT9 l, " dc N2 HH ^`T ?Oval 63 pcD<4___PPT9 l, " dc N2 IH ^?Oval 64q8  D<4___PPT9 l, " dc N2 JH ^q?Oval 65$8 ~ D<4___PPT9 l, " dc N2 KH ^@?Oval 66 8  D<4___PPT9 l, " dc N2 LH ^ q?Oval 67j 8 D<4___PPT9 l, " dc N2 MH ^ ?Oval 688 m D<4___PPT9 l, " dc vB NHB F?Line 69G 9 > H H0 ?Text Box 70 Fs D<4___PPT9 zTree 1, " d2c 9 H HP ?Text Box 72  D<4___PPT9 u1, " d2c 9 H HX ?Text Box 73 h D<4___PPT9 u2, " d2c 9 H H@Q ?Text Box 74 @ D<4___PPT9 u3, " d2c 9  H H`Y ?Text Box 75p X x W D<4___PPT9 u4, " d2c C  H DQ ?eW[eJX 1 p D<4___PPT9 5, " dc  C  H DZ ?eW[eJX 2  D<4___PPT9 7, " dc  C  H D[ ?eW[eJX 3~ D<4___PPT9 6, " dc  C  H DV ?eW[eJX 4j D<4___PPT9 8, " dc  C H D X ?eW[eJX 5 D<4___PPT9 9, " dc   C7"IDqIi+z2O___PPT10i.LwM+D=' = @B +"U! TT0EELT(  L% L  x 0e0e?Rectangle 2,YS   YAn example: m = 5  Z L  x@;0e0e?Rectangle 3bb >6___PPT9 H n"Px ]m  LC *Group 71"]m L2 L \ x(?Oval 5]qD<4___PPT9 l, " dc L2 L \ ?Oval 6qjD<4___PPT9 l, " dc L2 L \`~(?Oval 7 q D<4___PPT9 l, " dc L2  L \z(?Oval 8V q D<4___PPT9 l, " dc L2  L \:?Oval 9qYD<4___PPT9 l, " dc N2  L ^?Oval 10^pD<4___PPT9 l, " dc N2  L ^?Oval 11pkD<4___PPT9 l, " dc N2  L ^ ?Oval 12 p D<4___PPT9 l, " dc N2 L ^?Oval 13W p D<4___PPT9 l, " dc N2 L ^?Oval 14pZD<4___PPT9 l, " dc |B L LԔ?Line 15a  L # Z@ABCDEFQRUVWXԔ?Pu+e@  Freeform 16aN2 L ^ ?Oval 17g9  D<4___PPT9 l, " dc N2 L ^`?Oval 189 t D<4___PPT9 l, " dc N2 L ^?Oval 19 9  D<4___PPT9 l, " dc N2 L ^?Oval 20` 9 D<4___PPT9 l, " dc N2 L ^?Oval 21 9 c D<4___PPT9 l, " dc  L # Z@ABCDEFQRUVWXԔ?b1zN" N@  bNFreeform 22jf ` |B L LԔ?Line 23f f |B L LԔ?Line 24==9 N2 L ^@?Oval 25]pD<4___PPT9 l, " dc N2 L ^@v(?Oval 26pjD<4___PPT9 l, " dc N2 L ^@ ?Oval 27 p D<4___PPT9 l, " dc N2 L ^?Oval 28V p D<4___PPT9 l, " dc N2 L ^@?Oval 29pYD<4___PPT9 l, " dc vB L F?Line 30O ] N2  L ^ ?Oval 31g8  D<4___PPT9 l, " dc N2 !L ^ࡴ?Oval 328 t D<4___PPT9 l, " dc N2 "L ^?Oval 33 8  D<4___PPT9 l, " dc N2 #L ^?Oval 34` 8 D<4___PPT9 l, " dc N2 $L ^ ?Oval 35 8 c D<4___PPT9 l, " dc N2 %L ^ ?Oval 36]pD<4___PPT9 l, " dc N2 &L ^?Oval 37pjD<4___PPT9 l, " dc N2 'L ^?Oval 38 p D<4___PPT9 l, " dc N2 (L ^`?Oval 39V p D<4___PPT9 l, " dc N2 )L ^`?Oval 40pYD<4___PPT9 l, " dc N2 *L ^?Oval 41g8  D<4___PPT9 l, " dc N2 +L ^?Oval 428 t D<4___PPT9 l, " dc N2 ,L ^`?Oval 43 8  D<4___PPT9 l, " dc N2 -L ^?Oval 44` 8 D<4___PPT9 l, " dc N2 .L ^ ?Oval 45 8 c D<4___PPT9 l, " dc vB /L F?Line 46 ] N2 0L ^v(?Oval 47]pD<4___PPT9 l, " dc N2 1L ^` ?Oval 48pjD<4___PPT9 l, " dc N2 2L ^ ?Oval 49 p D<4___PPT9 l, " dc N2 3L ^@?Oval 50V p D<4___PPT9 l, " dc N2 4L ^?Oval 51pYD<4___PPT9 l, " dc N2 5L ^?Oval 52g8  D<4___PPT9 l, " dc N2 6L ^@?Oval 538 t D<4___PPT9 l, " dc N2 7L ^ ?Oval 54 8  D<4___PPT9 l, " dc N2 8L ^`?Oval 55` 8 D<4___PPT9 l, " dc N2 9L ^?Oval 56 8 c D<4___PPT9 l, " dc vB :L F?Line 57= L N2 ;L ^?Oval 58gpD<4___PPT9 l, " dc N2 L ^?Oval 61` p D<4___PPT9 l, " dc N2 ?L ^?Oval 62 pcD<4___PPT9 l, " dc N2 @L ^?Oval 63q8  D<4___PPT9 l, " dc N2 AL ^?Oval 64$8 ~ D<4___PPT9 l, " dc N2 BL ^?Oval 65 8  D<4___PPT9 l, " dc N2 CL ^@?Oval 66j 8 D<4___PPT9 l, " dc N2 DL ^@?Oval 678 m D<4___PPT9 l, " dc vB ELB F?Line 68 x 9 > L H`?Text Box 70 Fs D<4___PPT9 zTree 2, " d2c  C7"IDqIi+z2O___PPT10i.Nw0E+D=' = @B +"! @P(  P; P  j 0e0e? jL 1,YS   } On K2m+1 Case0 (     P  z 0e0e?gQ[HrbMn@S 2" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!p3=>drs/shapexml.xml[Oo6wx-\۲8F mQuZmQtKНa]va@.;A~$󀡱ȑH>DnRQ[Wʖ3ZӣA{x KcJCd! W972~^k$;{.j\"'q >1={=HQ/(-OԗDdzU+*A6/'v eZbPW$|`MOUc5 ?BF9@m+(x8 u/3H-R?7Uz<%РrQA0dU027挅I3*dIap)6IBBuSQq<0@iiC[Aڮٜ4>e@԰7ZFyL 3qMdjTfB0\Ij+k >~@WŔ6ǮTuPM?]$%?FhCl~9gf"irL.2,4#Wp69{=N_>hd &a=,|\tmH|U| _5 \?߽ `!\Yua~̅0daGW"i#96^47,ki9#وRo0aUT!76o|V9Iq=VӼ^@JIHmz{*N—^ kl,3y./iu(EPABI鋊 mc`ѕR;#42M[iPaK(ln0x{oI7X*u}uzz~(>_ԫV.?Zr _ g?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!p3=>)drs/shapexml.xmlPK-!Tl-Sdrs/downrev.xmlPK]'<  Constantine s Weak Conjecture on odd order (2005) For any m e" 2, K2m+1 can be (2m+1)-edge colored in such a way that the edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs. Later, we (with Y. H. Lo) verify this conjecture by decomposing K2m+1 (with designed coloring) into multicolored Hamilton cycles. All 2m+1 colors occur in a Hamilton cycle. 2&$& "&...R&&& &&&6& &&...+&& &+&&  Q P J`?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.8PœA+D=' = @B +"! PT(  T T  j0e0e? jL 1,YS   >   T  z 0e0e?gQ[HrbMn@S 2Y  x If the edge-coloring is not designed, then finding multicolored spanning trees is not that easy except the first one!yy"$x  Q T J?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc b T :A?? WGr 1 Z" C7"IDqIi+z2O___PPT10i.:Z5Ǫ+D=' = @B +"! `X(  XQ X J ?bq_Gr}_HrbMn@S 5^ `M<4___PPT9  *,(" dc } X  x`0e0e?Rectangle 2"  PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!eNFdrs/shapexml.xml\n6?<~I# 궀10 h5('UQEiEI4(+BER^>|ndΥRѡ5JQ$[iQ4#w-WgmTuD]`ަ S0W3i4FvPMX$ ^%惬/]~ޗ$uhupO` ǜ4h5/0Kʅa_"HD?Jt6 ZR7F7MDsnYr3&C5%\#ez&=1S%J_.̈E%}:lSY&Kb#*,Պ> 3#uaϡ%<$ʲ}LɈl8!0j׮g=q&~ }j 2bmX^g"@#5hވh2g0:o8òEuP?,|%y yc8NIi.p6[x?! +HEI`eIN R0&>Pᨨ 0ֆ/2kƞ,cdx v( XlnRn5P¨6\FS-ҁd%pGp0/eG˜8?٤ABuחiQ$5\k0*1gͦ'0Wv8^bmHhd}`c8POmaڬ'.93ގ $+v*l#?~ X\AX8uQM"*2R: ? n$d23t|{F  _ ,mϕ,fIJ 1xS H>u}u1afˮB)՚)t̳գCтIW5бen 3[1}ylD湗 Kg,zWeZ/ս4i6;U`1@CEk^5;Wwްƻ49. j><~G[~M-s ȱ+%p2SPU[`v{|}b?=,#xیtJGJ݆eUodKڐe 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"&&$&&& & &&&&&..$&&&&&&1&&&:&g*S     X  ~60e0e?Rectangle 3Z   Multicolored Spanning Treesve aeaea  C7"IDqIi+z2O___PPT10i. ]+D=' = @B +">! E=p\%(  \Q \ J ?bq_Gr}_HrbMn@S 5^ `M<4___PPT9  *,(" dc _ \  x`40e0e?Rectangle 2" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!s?oLgdrs/shapexml.xml]n6wx[F" mQu^Lj$JCjŞb!e[ڮK 6)xeBQ*!tɳ>{}:uXk.k7wĒ>e:x )]4L ry͠WZ ĤnS(,$yWeूO}Ϗ~@ CiyDœ @`/x\ETZP .}@pw>u6Y]([bJ\H *o%JLovm1~NjW/8 ,b7Y$8,v\\rOKw}M_\ٺ~ XLV諢_<<ce7IonVa"1F72[9g'&dͿFWC>,+O[jnGgMAqiZVp I 7Hda<)Ygt=xOG?gI=xYgA{w=x{8tM՝q˾PiW5~ ˠt0_z!tZW qO,"F/s=N53G֣7ktxa"_fi|uVg4^,?~ޖi8m) G T-}jm hY|2sg,^wᵵ@nNrฏP="Uvºju[pܬ=vȭq}QE^t3./a*zG5t:rGQ; ;ϕhwq{;BNG\qDA;Aۯ)lkQ~ـh#K7aw5ROۄR#u)/y[.-ZCna>{|b.tنE9 hBr}ՐF?bt5[&ɄkT&>4GCT{Q-G/O篅O@}~W6R@M q};8% 􁜀m)PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!s?oLg)drs/shapexml.xmlPK-!:drs/downrev.xmlPK ';  4Constantine s Strong Conjecture (2002) If m > 2, then in any properly (2m 1)-edge-colored K2m, all edges can be partitioned into m isomorphic multicolored spanning trees. Constantine s Strong Conjecture on odd order (2005) In any properly (2m+1)-edge-colored K2m+1, all edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs. Unicyclic graph: a graph with exactly one cycle. x&&&&& && &&&&..$&&& &&&& &&&%&& &&"&&...$&&& &&&&&"&g8J   O  - \  ~30e0e?Rectangle 3Z'A    [ Conjectures e   C7"IDqIi+z2O___PPT10i. ]+D=' = @B +"! `(  ` `  j 0e0e? jL 7,YS   a Observation      `  z 0e0e?gQ[HrbMn@S 8Y  6The above two conjectures are too hard to be verified in my opinion. There is a chance to get something done on the conjecture posed by Brualdi and Hollingsworth. Find more (3) multicolored spanning trees!> %$$$8    Q ` J ?bq_Gr}_HrbMn@S 4^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.YPߺ+D=' = @B +"! dl(  d% d  jp(0e0e? jL 1,YS   gRecent progress"$   d  z`90e0e?gQ[HrbMn@S 2Y  ^By using probabilistic techniques, Carraher et al. prove that about m/(500log(2m)) mutually edge-disjoint rainbow spanning trees exist for m e" 500,000. We construct at least [(6m+9)1/2/3] such trees by using an explicit construction. This is a joint work with Yuan-Hsun Lo, K. E. Perry and C. A. Rodger.B0D $b (x /  Q d J@q(?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.*+D=' = @B +"! h(  h* h  j 0e0e? jL 1,YS   lAlgorithmic approach"$   h  z0e0e?gQ[HrbMn@S 2D  FThe first tree we can get is a spanning star with the center called root. In order to find the second one, we have to make a choice of the second root. Clearly, we have to make an adjustment of the first tree. Otherwise, the second tree will not be a spanning tree. So, to search for the (k+1)th tree we adjust the previous all k trees. We stop when there is no way to adjust previous ones. (Our work depends on the number of pendant leaves.)L" &"*"  Q h J ?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.*γ+D=' = @B +"*+! &&l&(  l0 l  ~0e0e?Rectangle 3,YS   ^Edge-switchinge $ l  x 0e0e?Rectangle 2" ì PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Adrs/shapexml.xml]n8/ :p'N;1Z$104ؠeVvsZ,2y[GJl4]bTcSz:ѝeߴX$TOS5AEjeDEɾ_X"f c~Y& M^w\g BR( djmfGw= \Ys=뜟9Tɐնեk=g:xl?=흜*X(yǢvmYĵZ7+ΦF1 :*,9zNJk"w?owzdZe׹tq טU!NfEb m[ ]>"|pnX. V`3{?%G|2~- TM1k\"Brn|枧6h3b\K]R_R)U1e.ө|Q =fZ qfS?Qw٧4UW /?>C IL54\|b#_>LH&-E^S|#P~A1 HSs~'{}ğ#~3a0QX&E 4=!$u7/ÇKkk~#ܗrjҍf\MJ} a$>}OɈ \GÍVp+:ėA _B 8 ?m=~GBu!!Y/YvTQREq9SKQ?dףVJ`wgk3_^F5}wġУ<I-P?ih"h.< ƒI= %)3!EKjuJmS<,8eq0\Hh=Z(j_P@ÅQ/%E&pE(pt6Jjf݃Cj /A :hS| L =R2 $b̡(dIeteHֱ2O?fMlV9>Riz @Fs , -АG"t"\} ~+k2֮e*W2{fq`nfZ. P߾p=؃@~{>{ c^/%Ik_8Ie:$lHv!VCB?hLd\B܁Qb(xIJLڌ]q~$i+/vmº+u&U1[SҶoB(:A=oN5y&Vm3b.21;vαjC7wDj*Y㑂˹dTndvkkS|ҥ\$Zu}jV*A!-W+EbW(14mlR\sZ4rG\$[˙yNK.d ZT- d5~ԱwZ qdQd1Y*|͢du)Nc8'ҸσoF/FBMG\sDGUf4fXݘ x?ff¯++U.qRcĢ@4 /6TE⢔m2}? ȿ[vBg;Fn}ʧ,.PK!sdrs/downrev.xmlDN0EH5Hh u+*MibǑ)_;:٢ؓƱ A\9mV.S!"kl#XOOfkwXᐣ&.2T Y lŘN_Kඕ, ;oZ(3{Ǘ}<(u~݂W._ԳV0N.q.0D _M ?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!A)drs/shapexml.xmlPK-!s drs/downrev.xmlPK   : 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.) 2$$$$$$$$,,$$+$$$$$$,$$,$$,$$, $$$$$$,$$,$$,$$,$$$$,$$,$$,$,$$,$, $$,$$,$$,$$,$$,$,$$,$, $$  l l1?Text Box 31"k e PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!`V[drs/shapexml.xml\nF?Hz"rڅ,EY!KlE?tE)Eΐ(qWԌ Ew3$=_DqY: ',7o|~7ϒ^1zIߜ j̛˞Ɖ4N#&1%\0 EanۭZn`*~=JR:^\F0M,%0pZ>bS0H3,}H`+<>1~i|3$+5)\!'RO 5X cөvmPnu!둒<|l9f0imіIBC42h+eLRxcU NH \y`AkwO;rS˵ONkNy89>>q~&9noL&>q߳(8G5C܆O]! MGq,DeXmU "֒+XSwz:mM kl{mٰǓꒆ ~2c6ُ?J[J֋FDr둣!n.G3Ydi*$, πRaj2PxFqlrKV?;IГCA31a{f&v rС!?J&z6bJ#J=/$OC@y8@1R~xn'rƊd=b+sߓaroz5"|7_+BeND\Iri$LOsN-y-;a00+8I P8E8&UU"7Z Y!VيATUAV)hBa!:'1?hJy}5=?4q#L+Q%~;F2E ŭKPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!`V[)drs/shapexml.xmlPK-!PW drs/downrev.xmlPKZ ) p$<4___PPT9 Fact: If (e1) = (y2z2) and (e2) = (y1z1), then T[x; y1, y2; z1, z2] is multicolored.[ " d2&&&&.&&.&.&&.&&.&. &&&&&&.&&.&&.&&.&g[  C7"IDqIi+z2O___PPT10.`;Lp+PQFDi' = @B D$' = @BA?%,( < +O%,( < +D[' = @B%(D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(+8+0+l +" j5! //%%p/(  pQ p J?bq_Gr}_HrbMn@S 6^ `M<4___PPT9  *,(" dc % p  ~0e0e?Rectangle 2,YS   SMSTe \2 p  t1?Oval 5PD<4___PPT9 l, " dc \2 p  t 1?Oval 6% 7 D<4___PPT9 l, " dc \2 p  t@1?Oval 7 ) D<4___PPT9 l, " dc \2 p  t@1?Oval 8 t) D<4___PPT9 l, " dc \2 p  t1?Oval 9% W D<4___PPT9 l, " dc ^2  p  v@1?Oval 10% < D<4___PPT9 l, " dc ^2  p  v1?Oval 11 ID<4___PPT9 l, " dc ^2  p  v঴1?Oval 12% I D<4___PPT9 l, " dc ^2  p  v 1?Oval 13% ] D<4___PPT9 l, " dc 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 20j3 ] p l1?Text Box 22]:w}D<4___PPT9 ux, " d2c  p l1?Text Box 23gD<4___PPT9 e1R " d2ckc  p l 1?Text Box 24}k D<4___PPT9 e2R " d2ckc  p l1?Text Box 25S ,s D<4___PPT9 y1R " d2ckc  p l`1?Text Box 26S  @ s D<4___PPT9 y2R " d2ckc  p lx(1?Text Box 27 W D<4___PPT9 z1R " d2ckc  p l71?Text Box 28S  s D<4___PPT9 z2R " d2ckc C }  pC *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 "p  p1?Line 19}w% B #p  p1?Line 21} S B $pB  p1?Line 29S w B %p  p1?Line 30jS S v  5  pC *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} ,$DB p  p8c?Line 32_B p  p8c?Line 33B  pB  p8c?Line 34 5 B !p  p8c?Line 35 p l 81?Text Box 38P  D<4___PPT9 T[x; y1, y2; z1, z2] " d2gggggoggoggoggog   p  v0e0e?eW[HrbMn@S 39   @"   C7"IDqIi+z2O:2___PPT10.`;Lp+ D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(Ds' = @B%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(+" b! iatI(  t* t  j0e0e? jL 5,YS   lCan we add one more?",   t  zu(0e0e?gQ[HrbMn@S 6Y  4In order to find a new one, we first  revise the multicolored spanning trees which have been obtained earlier and then define a new one accordingly. This depends on the number of pendant leaves which occurred in previous trees. The argument for trueness is somehow very complicate.D $M  $   Q t J?bq_Gr}_HrbMn@S 4^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.YI`+D=' = @B +"s! x(  xt x  j0e0e? jL 5,YSj  F>___PPT9  h Quick note     x  z0e0e?gQ[HrbMn@S 6Yj F>___PPT9  z If we have k edge-disjoint spanning trees, then the maximum degree of each tree is at most 2m - k. In our construction, this value is 2m  2k since we use edge-switching. Hence, our algorithm can only go this far. A smarter idea is need to improve this result.0   a x J`?bq_Gr}_HrbMn@S 4^ `WF>___PPT9   *&""    C7"IDqIi+z2O80___PPT10.i+6"! |(  |< |  jq(0e0e? jL 5,YS   ~For K2m+1 case0 (   * |  z`y(0e0e?gQ[HrbMn@S 6Y  \By a similar argument with one pendant vertex fixed, we are able to find around the same number of multicolored spanning unicyclic subgraphs. Clearly, there is room for improvement; O(m1/2) is still far from m.J $,$   Q | J8?bq_Gr}_HrbMn@S 4^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.Yd<+D=' = @B +"! (  C   j1 0e0e? jL 1,YS   )Smaller multicolored isomorphic subgraphs**& )     zu0e0e?gQ[HrbMn@S 2Y  0It is easy to see that if Kn (with designed or un-designed edge coloring) can be decomposed into C3 s, then all C3 s are multicolored. But, it is not that trivial for C4-decomposition even we can design the edge coloring. We expect all cycle decompositions of admissible order and a designed edge coloring, a multicolored cycle design can be obtained.(?) This is what I am working now: Rainbow graph designs."*F"*"*6"*"&3"&  Q  J5?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.8F "+D=' = @B +"-! ))"")(  R   j50e0e? jL 1,YS   Examples (Designed coloring)8 "$&$"$  2   z`0e0e?gQ[HrbMn@S 2qYO  dtA 4-cycle design of order n exists if and only if n a" 1 (mod 8). For each odd integer n, there exists an n-edge coloring (Class 2) of Kn. The coloring can be designed as follows. Let V(Kn) = Zn and the edge {x, y} receives the color  i if x + y is congruent to 2i modulo n. (We use color n for 0.) 4"Sck7ckcksca :  /w : @  C *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 ,$DR   b?AutoShape 40 j D<4___PPT9 l, " dc R   b?AutoShape 41: ) D<4___PPT9 l, " dc R   b?AutoShape 42 @ ) D<4___PPT9 l, " dc R   b?AutoShape 43G)D<4___PPT9 l, " dc R  b ?AutoShape 44 a D<4___PPT9 l, " dc vB B FjJ?Line 45g = vB  FjJ?Line 46g wvB  FjJ?Line 47w0 vB  FjJ?Line 480  vB  FjJ?Line 49=  vB B FjJ?Line 50w =vB  FjJ?Line 51= 0 vB  FjJ?Line 52g 0 vB  FjJ?Line 53g  vB  FjJ?Line 54w  E  H?Text Box 55D<4___PPT9 y10 " d2g E  H`?Text Box 56U ) D<4___PPT9 y10 " d2g E  H ?Text Box 57-  D<4___PPT9 y20 " d2g E  H`?Text Box 58  : D<4___PPT9 y20 " d2g E  H ?Text Box 59$  D<4___PPT9 y30 " d2g E  Hx ?Text Box 60M 6  D<4___PPT9 y30 " d2g E  H5?Text Box 61j D<4___PPT9 y40 " d2g E   H0?Text Box 62OH  D<4___PPT9 y40 " d2g E ! H5?Text Box 63 D<4___PPT9 y50 " d2g E " H0?Text Box 64:  D<4___PPT9 y50 " d2g I  F@ ?eW[eJX 30 8 D<4___PPT9 10 " dg  I  F?eW[eJX 31\   D<4___PPT9 20 " dg  I  F  ?eW[eJX 32 v D<4___PPT9 50 " dg  I  F@?eW[eJX 33G jD<4___PPT9 30 " dg  I  F@?eW[eJX 34Y|D<4___PPT9 40 " dg   C7"IDqIi+z2O___PPT10. >0}W+,D|' = @B D7' = @BA?%,( < +O%,( < +Dn' = @B%(%(D' = @B%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+"f! meM(  8   j0e0e? jL 1,YS   z"Cyclic multicolored 4-cycle system##" "  |   zr(0e0e?gQ[HrbMn@S 2Y  ZLet n = 8k + 1, k is a positive integer. The base cycles are: (3, 4i+1, 4, 4i+3), i = 1, 2, & , k. It is not difficult to check each difference from 1 to 4k occurs exactly once and the colors in the cycle (3, 4i+1, 4, 4i+3) are 2i + 2, 2i + 3, 4k - 2i - 2 and 4k - 2i  3 and they are distinct... -  Q  J?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.">+D=' = @B +"a! h` H(  *   j 0e0e? jL 1,YS   lOther cycle-systems?"$     z00e0e?gQ[HrbMn@S 2Y  #As we have seen multicolored cycle systems of order n do exist when the cycle lengths are 4 and n. Problem: Find a multicolored cycle system for each cycle length larger than 4. Example: For the same edge-coloring, (0, 3, 8, 10, 11, 4) generates a multicolored 6-cycle system of order 13.T$c $ F$ $i #  Q  J`?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.>+D=' = @B +"^! e]0E(  X   j0e0e? jL 1,S   &Follow up (Arbitrarily given coloring)4'  &  &  T   z =0e0e?gQ[HrbMn@S 2Y  4Can we find a multicolored fairly long path in a properly colored complete graph? Given an n-edge-colored Kn, a rainbow Hamiltonian path exists for n > n0. (!?) (I believe that this is a true.) Problem: How many edge-disjoint rainbow matchings of fixed size can we find in an edge-colored complete graph?  5R$ $$$$, $$ $$$,$$$ $f$$$5 Q  J?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i. /+D=' = @B +"U! \T@<(  )   j@10e0e? jL 1,YS   kMulticolored 4-cycles   z   z0e0e?gQ[HrbMn@S 2Y  2Given a complete graph of order n = 8k+1 which is n-edge colored. Can we decompose the graph into k(8k+1) multicolored 4-cycles? Clearly, we have no idea about how the edges are colored, only a proper coloring is known. We should be able to do it in my opinion, but I am not able to get this job done yet!:3 $ V$2  Q  J;?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.I@+D=' = @B +"Ґ! ُяP(  W   j0e0e? jL 1,S   #Multicolored 4-cycle design of Kn,n6$ "$*$"$#  (   z0e0e?gQ[HrbMn@S 2Y  ZIf n is even and Kn,n is n-edge-colored, then there exists a multicolored design of Kn,n provided n e" 4. This fact is equivalent to partition a Latin square of order n into n2/4 sub-squares. 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A multicolored matching is then a partial transversal of the square. Everyone knows that finding a partial transversal of maximum size in an arbitrary Latin square is a very difficult problem. Ryser believed the size is n if n is odd and Brualdi conjectured n - 1 in general. How about smaller matchings?Py"*"S&&x  Q  J ?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.P[ӷ+D=' = @B +"! p(  6   j@q0e0e? jL 1,Y   xProblems I would like to work!$     z@ 0e0e?gQ[HrbMn@S 2f B:___PPT9 Prove that an (8k+1)-edge-colored K8k+1 can be decomposed into multicolored 4-cycles, i.e. a multicolored 4-cycle design exists for each admissible order. Extend the above result to 4-cycle GDD. How about multicolored k-cycle designs for admissible orders? jn#"*" *"*"  DQ  Jq?bq_Gr}_HrbMn@S 3^ `M<4___PPT9  *,(" dc  C7"IDqIi+z2O___PPT10i.I!+D=' = @B +"! 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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. A. 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. Comput. 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. 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L. Fu and Y. H. Lo, Multicolored parallelisms of Hamiltonian cycles, Discrete Math. 309 (2009), No. 14, 4871-4876. [6] H. L. Fu and Y. H. Lo, Multicolored isomorphic spanning trees in complete graphs, Ars Combinatoria, 122 (2015), 423 - 430. [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. L. Fu, Yuan-Hsun Lo, K. E. Perry and C. A. Rodger, On the number of rainbow spanning trees in edge-colored complete graphs, in preprints. . .x"&y"&~"&R""**""** "&&&"g  C7"IDqIi+z2O___PPT10i. ~+D=' = @B +"! 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