Edge-disjoint rainbow spanning trees in complete graphs

Transcription

1 Edge-disjoi raibow spaig rees i complee graphs James M arraher Sephe G Harke Paul Hor April 30, 013 Absrac Le G be a edge-colored copy of K, where each color appears o a mos / edges he edgecolorig is o ecessarily proper A raibow spaig ree is a spaig ree of G where each edge has a differe color Brualdi ad Holligsworh [4] cojecured ha every properly edge-colored K 6 ad eve usig exacly 1 colors has / edge-disjoi raibow spaig rees, ad hey proved here are a leas wo edge-disjoi raibow spaig rees Kaeko, Kao, ad Suzuki [13] sregheed he cojecure o iclude ay proper edge-colorig of K, ad hey proved here are a leas hree edgedisjoi raibow spaig rees Akbari ad Alipouri [1] showed ha each K ha is edge-colored such ha o color appears more ha / imes coais a leas wo raibow spaig rees We prove ha if 1, 000, 000 he a edge-colored K, where each color appears o a mos / edges, coais a leas /1000 log edge-disjoi raibow spaig rees Keywords: raibow spaig rees AMS classificaio: Primary: 0515; Secodary: 0505, Iroducio Le G be a edge-colored copy of K, where each color appears o a mos / edges he edge-colorig is o ecessarily proper A raibow spaig ree is a spaig ree of G such ha each edge has a differe color Brualdi ad Holligsworh [4] cojecured ha every properly edge-colored K 6 ad eve where each color class is a perfec machig has a decomposiio of he edges of K io / edge-disjoi raibow spaig rees They proved here are a leas wo edge-disjoi raibow spaig rees i such a edge-colored K Kaeko, Kao, ad Suzuki [13] sregheed he cojecure o say ha for ay proper edge-colorig of K 6 coais a leas / edge-disjoi raibow spaig rees, ad hey proved here are a leas hree edge-disjoi raibow spaig rees Akbari ad Alipour [1] showed ha each K ha is a edge-colored such ha o color appears more ha / imes coais a leas wo raibow spaig rees Our mai resul is Theorem 1 Le G be a edge-colored copy of K, where each color appears o a mos / edges ad 1, 000, 000 The graph G coais a leas /1000 log edge-disjoi raibow spaig rees The sraegy of he proof of Theorem 1 is o radomly cosruc /1000 log edge-disjoi subgraphs of G such ha wih high probabiliy each subgraph has a raibow spaig ree This resul is he bes kow for he cojecure by Kaeko, Kao, ad Suzuki Hor [1] has show ha if he edge-colorig is a Deparme of Mahemaics, Uiversiy of Nebraska Licol, s-jcarrah1@mahuledu Parially suppored by NSF gra DMS Deparme of Mahemaics, Uiversiy of Nebraska Licol, harke@mahuledu Parially suppored by NSF gra DMS Deparme of Mahemaics, Harvard Uiversiy, phor@mahharvardedu 1

2 proper colorig where each color class is a perfec machig he here are a leas ɛ raibow spaig rees for some posiive cosa ɛ, which is he bes kow resul for he cojecure by Brualdi ad Holligsworh There have bee may resuls i fidig raibow subgraphs i edge-colored graphs; Kao ad Li [14] surveyed resuls ad cojecure o moochromaic ad raibow also called heerochromaic subgraphs of a edge-colored graph Relaed work icludes Brualdi ad Holligsworh [5] fidig raibow spaig rees ad foress i edge-colored complee biparie graphs, ad osaie [8] showig ha for cerai values of here exiss a proper colorig of K such ha he edges of K decompose io isomorphic raibow spaig rees The exisece of raibow cycles has also bee sudied Alber, Frieze, ad Reed [] showed ha for a edge-colored K where each color appears a mos c imes he here is a raibow hamiloia cycle if c < 1/64 Rue see [11] provided a correcio o he cosa Frieze ad Krivelevich [11] proved ha here exiss a c such ha if each color appears a mos c imes he here are raibow cycles of all leghs This paper is orgaized as follows Secio icludes defiiios ad resuls used hroughou he paper Secio 3, 4, ad 5 coais lemmas describig properies of he radom subgraphs we geerae The fial secio provides he proof of our mai resul Defiiios Firs we esablish some oaio ha we will use hroughou he paper Le G be a graph ad S V G Le G[S] deoe he iduced subgraph of G o he verex se S Le [S, S] G be he se of edges bewee S ad S i G For aural umbers q ad k, [q] represes he se {1,, q}, ad [q] k is he collecio of all k-subses of [q] Throughou he paper he logarihm fucio used has base e Oe iequaliy ha we will use ofe is he uio sum boud which saes ha for eves A 1,, A r ha [ r ] r P A i P [A i ] Throughou he res of he paper le G be a edge-colored copy of K, where he se of edges of each color has size a mos /, ad 1, 000, 000 We assume G is colored wih q colors, where 1 q Le j be he se of edges of color j i G Defie c j = j, ad wihou loss of geeraliy assume c 1 c c q Noe ha 1 c j / for all j Le = / log where = 1000 Noe ha we have o opimized he cosa, ad i ca be slighly improved a he cos of more calculaio Sice log 1 log we have 1 log log ad 1 log log We will frequely use hese bouds o We cosruc edge-disjoi subgraphs G 1,, G of G i he followig way: idepedely ad uiformly selec each edge of G o be i G i wih probabiliy 1/ Each G i cosidered as a ucolored graph is disribued as a Erdős-Réyi radom graph G, 1/ Noe ha he subgraphs are o idepede We will show ha wih high probabiliy each of he subgraphs G 1,, G simulaeously coai a raibow spaig ree To prove ha a graph has a raibow spaig ree we will use Theorem below ha gives ecessary ad sufficie codiios for he exisece of a raibow spaig ree Broersma ad Li [3] showed ha deermiig he larges raibow spaig fores of H ca be solved by applyig he Maroid Iersecio Theorem [10] see Schrijver [15, p 700], o he graphic maroid ad he pariio maroid o he edge se of H defied by he color classes Schrijver [15] raslaed he codiios of he Maroid Iersecio Theorem io ecessary ad sufficie codiios for he exisece of a raibow spaig ree Suzuki [16] ad arraher ad Harke [6] gave graph-heoreical proofs of his same heorem

3 Theorem A graph G has a raibow spaig ree if ad oly if, for every pariio π of V G, a leas s 1 differe colors are represeed bewee he pars of π, where s is he umber of pars of π We show ha for every pariio π of V G io s pars, ha here are a leas s 1 colors bewee he pars for each G i Secios 3, 4 ad 5 describe properies of he subgraphs G 1,, G for cerai pariios π of V G io s pars May of our proofs use he followig varia of heroff s iequaliy [7], frequely aribued o Bersei see [9] Lemma 3 Bersei s Iequaliy Suppose X i are idepedely ideically disribued Beroulli radom variables, ad X = X i The λ P [X E[X] + λ] E[X] + λ/3 ad P [X E[X] λ] λ E[X] I several places i he paper we use Jese s iequaliy Lemma 4 Jese s Iequaliy see [17] Le fx be a real-valued covex fucio defied o a ierval I = [a, b] If x 1,, x I ad λ 1,, λ 0 wih λ i = 1, he f λ i x i λ i fx i We also make use of he followig upper bouds for biomial coefficies: e k = expk log k log k + k k k k 3 Pariios wih or 1 pars I his secio we show ha a pariio π of V G io or 1 pars has eough colors bewee he pars Sice color classes ca have small size, here migh o be ay edges of a give color i a subgraph G i Therefore, we group small color classes ogeher o form larger pseudocolor classes Recall ha c j is he size of he color class j, ad c 1 c c q Defie he pseudocolor classes D 1,, D 1 of G recursively as follows: l D k = where l is he smalles ieger such ha classes migh o coai all he edges of G j=1 l j=1 j j \ \ k 1 D i, k 1 D i /4 Noe ha he 1 pseudocolor Lemma 5 Each of he 1 pseudocolor classes D 1,, D 1 have size a leas /4 ad a mos / Proof osider he pseudocolor class D k, for 1 k 1 Sice each of he pseudocolor classes D 1, D k 1 has size a mos /, here are a leas k edges o i k 1 D i Therefore here exiss l ad l such ha D k = l i=l i, where D k = l i=l c i /4 If l = l he D k = l / Oherwise, we kow c l c l 1 c l /4 So, D k = l 1 i=l c i + c l 4 + c l =, which proves ha he pseudocolor class D k has size a mos 3

4 Lemma 6 For a fixed subgraph G i ad pseudocolor class D j, [ P EG i D j D j 3 ] log 1 3 As a cosequece, wih probabiliy a leas 1 1 every subgraph G i has a leas oe edge from each of he pseudocolor classes D 1,, D 1 Proof Fix a subgraph G i ad a pseudocolor class D j The expeced umber of edges i G i from he pseudocolor class is Dj By Bersei s iequaliy where λ = 3 log, we have [ P EG i D j D j 3 ] 3 log log Dj 3 log = 1 3 Sice D j /4, 1, 000, ad 50, D j 3 log 4 3 log 1 The secod saeme follows from he previous iequaliies by usig he uio sum boud for he 1 pseudocolor classes ad subgraphs ad recallig ha < Lemma 6 shows ha if we cosider a pariio π of V G io s pars, where s = here mus be a leas 1 colors i G i bewee he pars of π I he case whe he pariio has s = 1 pars here is a mos oe edge iside he pars of π, so here are a leas colors i G i bewee he pars of π 4 Pariios where 1 14 s I his secio we cosider pariios π of V G io s pars where 1 14 s Firs, we iroduce a ew fucio ha will help wih our calculaios The fucio f will be used o boud he probabiliy ha q s colors do o appear bewee he pars of π i G i Lemma 7 For a ieger l ad real umbers c 1,, c q, defie fc 1,, c q ; l = exp 1 If 1 c j for each j, q c j =, ad I [q] q l l 4, he fc 1,, c q ; l Proof For coveiece we defie wi = j I c j for a subse I [q] laim 1 j I c j 49 l log 00 fc 1,, c q ; l f 1, 1,, 1, x, }{{},, ; l, }{{ } k 1 imes q k imes where 1 x <, ad where k ad x are so ha k 1 + q k + x = 4

5 Proof of laim 1 Sice fc 1,, c q ; l is a symmeric fucio i he c j s, i suffices o show ha whe c c 1, fc 1, c,, c q ; l fc 1 ɛ, c + ɛ,, c q ; l, where ɛ = mi{c 1 1, c } fc 1, c,, c q ; l = I [q]\{1,} q l + I [q]\{1,} q l 1 exp wi + I [q]\{1,} q l exp c 1 wi + exp exp c 1 c wi c wi The firs wo summaios are uchaged i fc 1 ɛ, c + ɛ,, c q ; l, ad hece i suffices o show ha for every I [q]\{1,} l 1, exp c 1 wi + exp c wi c 1 ɛ wi + exp c + ɛ wi This follows immediaely by Jese s iequaliy ad he covexiy of expαx + β as a fucio i x laim where k + q k f 1, 1,, 1, x, }{{},, }{{ } k 1 imes q k imes ; l f1,, 1, }{{} k imes,, }{{} q k imes Proof of laim The fucio f is decreasig i each c j, ad i paricular c k ; l, Now cosider f1,, 1, }{{},, }{{ } k imes q k imes ; l = I [q] q l exp 1 wi mi{l,k} r=max{0,l q k} mi{l,k} r=max{0,l q k} mi{l,k} r=max{0,l q k} mi{l,k} r=max{0,l q k} exp log k r q k l r exp 1 l r r k r q k q k l r exp 1 l r r exp q k l + r log 1 l + r r exp log q k l + r r l + r + l 1 + r + + by 5

6 log l Sice l 4 ad 50, we have l l Thus he sum above is bouded by exp 49 l log 00 Lemma 8 Le Π be he se of pariios of V G io s pars, where 1 14 s For a pariio π Π, le B π,i be he eve ha here are less ha s 1 colors bewee he pars of π i G i The [ ] P B π,i 1 π Π Proof Fix a subgraph G i ad a pariio π Π Recall ha 1,, q are he color classes of G wih sizes c 1,, c q, respecively Le I π,i be he se of colors ha do o appear o edges of G i bewee he pars of π The oal umber of edges i G ha have a color idexed by I π,i is i I π,i c j By covexiy of x, here are a mos s+1 edges iside he pars of π Noe ha if Iπ,i does o have size q s, he i coais a se I I π,i of size q s, ad he eve ha o edges of G i bewee he pars of π have colors i I π,i is coaied i he eve ha o edges of G i bewee he pars have colors i I Thus, Sice s P [B π,i ] [q] I q s 1 1 j I cj s+1 fc 1, c,, c q ; s 1 1 s+1 1 s + 1 fc 1, c,, c q ; s exp 49 s + 1 s log e 1 sice by Lemma , we kow s Thus we ca boud he previous lie by s + 1 s + 1 log log log 14 + by We ow perform a uio boud over all pariios π Π The umber of pariios of V G io s oempy pars is a mos s s s s = exp s log s + 1 log s s 6

7 Therefore, P π Π wih s pars B π,i s + 1 log log 14 + Sice = 1000 ad 1, 000, 000, we have log , ad sice s + 1 3, P π Π wih s pars B π,i 3 log = 1 3 This gives a boud o he probabiliy for a fixed pariio size s Usig he uio sum boud over all pariio sizes s, where 1 14 s, ad over all subgraphs complees he proof This proves whe s is large here are eough colors bewee he pars 5 Pariios where s 1 14 Nex, we prove several resuls ha will be used o show here are eough colors i G i bewee he pars of he pariio whe he umber of pars is small Our goal is o show ha for a pariio π of V G io s pars, he umber of edges bewee he pars i G i is so large ha here mus be a leas s 1 colors bewee he pars Lemma 9 For a fixed subgraph G i ad color j, P [ EG i j ] + 4 log 1 4 As a cosequece, wih probabiliy a leas 1 1, every color appears a mos + 4 log imes i every G i Proof Fix a color j ad a subgraph G i Order he edges of j as e 1,, e cj For 1 k c j, le X k be he idicaor radom variable for he eve e k EG i For a color class wih size less ha we iroduce dummy radom variables, so we ca apply Bersei s iequaliy For c j + 1 k /, le X k be a radom variable disribued idepedely as a Beroulli radom variable wih probabiliy 1/ By cosrucio, EG i j X = / k=1 X k ad E[X] = By Bersei s Iequaliy where λ = 4 log, we have P [ EG i j ] + 4 log P [X ] + 4 log 16 log log

8 = exp 16 log log log sice log, log 3 48/ sice 1, which proves he firs saeme The secod saeme follows from he previous iequaliy by usig he uio sum boud for he q color classes ad subgraphs, ad recallig ha q < ad < Lemma 10 Fix S V G Le B S,i be he eve The [ S, S ] S S G i P 6 S S mi{ S, S } log S V G B S,i 4 Proof Fix a subgraph G i ad a se of verices S V G Le r = S The expeced umber of edges i G i bewee S ad S is r r/ By Bersei s iequaliy wih λ = 6 r r mi{r, r} log, we have 6 r r mi{r, r} log P [B S,i ] = 3 mi{r, r} So P S V G B S,i / 3r + r r=1 r r / 3 r = r r=/ r=1 / r / r=1 r= 3r r Applyig he uio sum boud for he subgraphs gives he fial saeme of he lemma The previous lemma gives a lower boud o he umber of edges bewee S ad S We use his lemma o fid a lower boud o he umber of edges bewee he pars for a pariio π = {P 1,, P s } of V G Defiiio 11 For x [0, ], le fx = x x 6x x mi{x, x} log If oe of he bad eves B S,i from Lemma 10 occur, he he sum 1 π={p f P 1,,P s} i, where s P i =, is a lower boud o he umber of edges bewee he pars of he pariio π We boud his sum for all pariios If fx was covex he we could immediaely fid a lower boud by usig Jese s iequaliy 4 Sice fx is o covex, we boud i wih a fucio ha is covex 8

9 Le hx be a fucio wih domai [a, b] We say a fucio h is cocave dow if for x, y [a, b] ad λ [0, 1], he hλx + 1 λy λhx + 1 λhy Firs, we prese wo basic resuls abou cocave dow fucios Lemma 1 Le hx be a differeiable fucio wih domai [a, b] Suppose ha h is cocave dow o [z, b], where z a, b Le lx be he lie age o h a he poi z, hz The he fucio { lx if a x z, h 1 x = hx if z < x b is cocave dow Proof Le y 1, y [a, b] where y 1 y, ad λ [0, 1] If y 1 ad y are boh i [a, z] or [z, b] he sice l ad h are boh cocave dow h 1 λy λy λh 1 y λh 1 y, osider he case whe y 1 [a, z ad y z, b] Le λ [0, 1] ad w = λy λy Le b be he y-iercep of he lie lx, ie lx = h zx + b Sice h is cocave dow o he ierval [z, b], we kow ha h lies below he age lie lx o he ierval [z, b] I paricular, h zy + b hy Le ɛ = h zy + b hy 0 If w z, he we wa o show ha h 1 w = h zw y 1 + ly 1 hy ly1 y y 1 w y 1 + ly 1 Noe ha i is eough o show ha h z hy ly1 y y 1 Sice ɛ 0, we have h z h z y y 1 ɛ y y 1 = h zy + b ɛ h zy 1 + b y y 1 = hy ly 1 y y 1 Suppose w > z The lie bewee z ad y is give by hy hz y z w z + hz, ad by he cocaviy of hx o [z, b] we kow h 1 w hy hz y z w z + hz We wa o show hy hz y z w z + hz hy ly 1 y y 1 Thus w z + hz I is eough o show ha hy hz y z ɛy 1 ɛz hy ly1 y y 1 We kow y 1 < z ad ɛ 0 h zy h zy z h zy 1 y + h zzy 1 ɛy + ɛy 1 h zy h zy z h zy 1 y + h zzy 1 ɛy + ɛz h zy ɛ h zz y y 1 h zy ɛ h zy 1 y z h zy + b ɛ h zz + b y z hy hz y z h zy + b ɛ h zy 1 + b y y 1 hy ly 1 y y 1 Lemma 13 If h 1 ad h are cocave dow fucios, he hx = mi{h 1 x, h x} is cocave dow Proof For every x, y ad λ [0, 1] we have hλx + 1 λy = mi{h 1 λx + 1 λy, h λx + 1 λy} λ mi{h 1 x, h x} + 1 λ mi{h 1 y, h y} = λhx + 1 λhx 9

10 We ex defie several fucios ha will lead o a cocave dow lower boud for he fucio f Defie o [0, ] he fucios x x 6 x f 1 p = x log, x x 6x f p = x log Noe ha { f1 x 0 x /, fx = f x / < x, Le lx = f xx / f / be he age lie of f x a he poi, 4 3 log Le c be he poi such ha f 1 x achieves is maximum value o he ierval [0, ] Defie { lx 0 x /, f 3 p = f x / < x ad f 4 x = By Lemma 1 he fucios f 3 ad f 4 are cocave dow { f1 x 0 x c, f 1 c c < x O he ierval [0, ] defie f 5 x = mi{f 3 x, f 4 x} The fucio f 5 x is cocave dow by Lemma 13, where fx f 5 x for all x [0, ] Figure 1 shows he fucios fx ad lx used o creae f 5 x Lemma 14 The sum s fx i, where s x i = ad x i 1 for all i, is bouded below by s fx i s 1f1 + f s + 1 Proof The proof is broke up io wo cases based o wheher s /, or s > / Whe s / he fucio fx f 5 x, so s fx i s f 5x i Sice he fucio f 5 x is cocave dow he sum s f 5x is miimized whe here is oe par of size s + 1 ad all he oher pars are of size 1 Sice s + 1 /, we have f 5 s + 1 = f s + 1 Noe ha l1 f 1 1, which implies f 5 1 = f1 Thus s fx i s f 5 x i s 1f f 5 s + 1 = s 1f1 + f s + 1 Whe s > /, we have x i / for all i Therefore fx i = f 1 x i for all i Sice f 1 x is cocave dow he sum is miimized whe oe pars has size s + 1 ad he res have size 1 Lemma 15 Le π be a pariio of he verices of G io s pars Suppose oe of he eves B S,i from Lemma 10 hold for all S V G ad 1 i The i each of he subgraphs G 1,, G, he umber of edges bewee he pars of π is a leas 1 s log + s + 1s 1 s 1 6 s + 1 log whe s /, ad 1 1 s log s + 1s 1 + s + 1 6s 1 log whe s > / 10

11 c lx fx 0 / Figure 1: The fucio fx, alog wih he lie lx Proof If oe of he eves B S,i hold he he sum 1 π={p fx where s 1,,P s} P i = is a lower boud o he umber of edges bewee he pars of π By Lemma 14 we kow his sum is bouded below by 1 s 1f1 + f s + 1 Lemma 16 Le π be a pariio of he verices of G io s pars, where s 1 14 Suppose oe of he eves B S,i from Lemma 10 hold for all S V G ad 1 i, ad every color appears i each G i a mos + 4 log imes as i Lemma 9 The i each of he subgraphs G 1,, G, he umber of colors bewee he pars of π is a leas s 1 Proof Suppose here exiss a subgraph G i ad a pariio π io s pars where here are a mos s colors bewee he pars i G i The by assumpio here are a mos s + 4 log edges i G i bewee he pars of π We will show ha he umber of edges bewee he pars of π ca o be his small, givig a coradicio Suppose < s 1 14 By Lemma 15 here are a leas 1 1 s log s + 1s 1 + s + 1 6s 1 log 11

12 edges i G i bewee he pars of π If π has a mos s colors i G i bewee he pars, he s + 4 log s log s log + s + 1 s 1 Rearragig we have s s log log + s log s 1 s We will give a upper boud o he lef side ad a lower boud o he righ side ha give a coradicio Sice s is a ieger ad / < s, we have s s 1 1 = 3 Therefore s s log log 6 log + s + 1 s 1 log 6 1 log + log s + 1 s 1 log log + log by Sice = 1000 ad 1, 000, 000, bouded above by log log 10 ad log Thus he erm above is log We ex boud he righ side By we have 1 log, ad sice s 1 14, so + s + 1 log + log s + 1 log + log 14 = log + 14 Whe = 1000 ad 1, 000, 000 we have + 14 > , which gives a coradicio So, here mus be a leas s 1 colors i G i bewee he pars of π whe / < s 1 14 Suppose s / By Lemma 15 here are a leas 1 1 s log s + 1s 1 + s 1 6 s + 1 log edges i G i bewee he pars of π If π has a mos s colors i G i bewee he pars he s + 4 log s log s s + 1 log 1

13 Rearragig we have s s log log + 6 s + 1 log s Usig 1 log log from, we have s s log log + 6 s + 1 log log s + 1 log + log Sice = 1000 ad 1, 000, 000, bouded above by log 10 ad log 101 Thus he erm above is log log Boudig he righ side usig 1 log + s + 1 s + 1 log + log from, ad s, we have log + log = log 3 Agai, whe = 1000 ad 1, 000, 000 we have 3 > which leads o a coradicio Thus, here mus be a leas s 1 colors i G i bewee he pars of π whe s 6 Mai Resul Theorem 1 Le G be a edge-colored copy of K, where each color appears o a mos / edges ad 1, 000, 000 The graph G coais a leas /1000 log edge-disjoi raibow spaig rees Proof Recall ha = / log where = 1000 We perform he radom experime of decomposig he edges of G io edge-disjoi subgraphs G i by idepedely ad uiformly selecig each edge of G o be i he subgraph G i wih probabiliy 1/ Wih probabiliy a leas 1 7 oe of he bad eves from Lemmas 6, 8, 9, ad 10 occur i ay of he subgraphs G i Heceforh le G 1,, G be fixed subgraphs where oe of hese bad eves occur We wa o show ha each G i has a raibow spaig ree By Theorem i is eough o show ha for every pariio π of V G io s pars, here are a leas s 1 differe colors appearig o he edges of G i bewee he pars of π By Lemma 6, every G i has a leas oe edge from each of he 1 pseudocolor classes Whe s = here mus be a leas 1 colors i G i bewee he pars of π Whe s = 1 here is a mos oe edge iside he pars of π, so here are a leas colors i G i bewee he pars of π If 1 14 s, he by Lemma 8 every pariio π of V G io s pars has a leas s 1 colors i G i bewee he pars, for every subgraph G 1,, G 13

14 Fially, we assume ha s 1 14 Whe s = 1 here are zero colors bewee he pars, so he 1 14 Sice Lemmas 9 ad 10 hold, by Lemma 16 codiio is vacuously rue So suppose s he umber of colors bewee he pars of π is a leas s 1 for every subgraph G 1,, G Therefore all of he subgraphs G 1,, G coai a raibow spaig ree, ad so G coais a leas = /1000 log edge-disjoi raibow spaig rees Ackowledgemes The auhors hak Douglas B Wes ad he Research Experiece for Graduae Sudes REGS a he Uiversiy of Illiois a Urbaa-hampaig for heir suppor ad hospialiy Fudig for he auhors visi i he summer 01 was provided by he UIU Deparme of Mahemaics hrough Naioal Sciece Foudaio gra DMS , EMSW1-MTP: Research Experiece for Graduae Sudes Refereces [1] S Akbari ad A Alipour Mulicolored rees i complee graphs J Graph Theory, 543:1 3, 007 [] Michael Alber, Ala Frieze, ad Bruce Reed ommes o: Mulicoloured Hamilo cycles [Elecro J ombi 1995, Research Paper 10, 13 pp elecroic; MR b:05058] Elecro J ombi, :Research Paper 10, omme 1, 1 HTML docume, 1995 [3] Hajo Broersma ad Xueliag Li Spaig rees wih may or few colors i edge-colored graphs Discuss Mah Graph Theory, 17:59 69, 1997 [4] Richard A Brualdi ad Susa Holligsworh Mulicolored rees i complee graphs J ombi Theory Ser B, 68: , 1996 [5] Richard A Brualdi ad Susa Holligsworh Mulicolored foress i complee biparie graphs Discree Mah, 401-3:39 45, 001 [6] James M arraher ad Sephe G Harke Euleria circuis wih o moochromaic rasiios Prepri, 01 [7] Herma heroff A measure of asympoic efficiecy for ess of a hypohesis based o he sum of observaios A Mah Saisics, 3: , 195 [8] Gregory M osaie Edge-disjoi isomorphic mulicolored rees ad cycles i complee graphs SIAM J Discree Mah, 183: , 004/05 [9] Devda P Dubhashi ad Alessadro Pacoesi oceraio of measure for he aalysis of radomized algorihms ambridge Uiversiy Press, ambridge, 009 [10] Jack Edmods Submodular fucios, maroids, ad cerai polyhedra I ombiaorial Srucures ad heir Applicaios Proc algary Iera of, algary, Ala, 1969, pages Gordo ad Breach, New York, 1970 [11] Ala Frieze ad Michael Krivelevich O raibow rees ad cycles Elecro J ombi, 151:Research paper 59, 9, 008 [1] Paul Hor Raibow spaig rees i complee graphs colored by machigs Prepri, 013 [13] A Kaeko, M Kao, ad K Suzuki Three edge disjoi mulicolored spaig rees i complee graphs Prepri,

15 [14] Mikio Kao ad Xueliag Li Moochromaic ad heerochromaic subgraphs i edge-colored graphs a survey Graphs ombi, 44:37 63, 008 [15] Alexader Schrijver ombiaorial opimizaio Polyhedra ad efficiecy Vol B, volume 4 of Algorihms ad ombiaorics Spriger-Verlag, Berli, 003 Maroids, rees, sable ses, hapers [16] Kazuhiro Suzuki A ecessary ad sufficie codiio for he exisece of a heerochromaic spaig ree i a graph Graphs ombi, :61 69, 006 [17] Roger Webser ovexiy Oxford Sciece Publicaios The laredo Press Oxford Uiversiy Press, New York,