Monochromatic Structures in Edge-coloured Graphs and Hypergraphs - A survey

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1 Moochromatic Structures i Edge-coloured Graphs ad Hypergraphs - A survey Shiya Fujita 1, Hery Liu 2, ad Colto Magat 3 1 Iteratioal College of Arts ad Scieces Yokohama City Uiversity 22-2, Seto, Kaazawa-ku Yokohama, , Japa fujita@yokohama-cu.ac.jp 2 Cetro de Matemática e Aplicações Faculdade de Ciêcias e Tecologia Uiversidade Nova de Lisboa Quita da Torre, Caparica, Portugal h.liu@fct.ul.pt hery.liu@catab.et 3 Departmet of Mathematical Scieces Georgia Souther Uiversity 65 Georgia Ave, Statesboro, GA 30460, USA cmagat@georgiasouther.edu 19 April, 2015 Abstract Give a graph whose edges are coloured, o how may vertices ca we fid a moochromatic subgraph of a certai type, such as a coected subgraph, or a cycle, or some type of tree? Also, how may such moochromatic subgraphs do we eed so that their vertex sets form either a partitio or a coverig of the vertices of the origial graph? What happes for the aalogous situatios for hypergraphs? I this survey, we shall review kow results ad cojectures regardig these questios. I most cases, the edge-coloured (hyper)graph is either complete, or o-complete but with a desity costrait such as havig fixed idepedece umber. For some problems, a restrictio may be imposed o the edge-colourig, such as whe it is a Gallai colourig (i.e. the edge-colourig does ot cotai a triagle with three distict colours). May examples of edge-colourigs will also be preseted, each oe either showig the sharpess of a result, or supportig a possible cojecture. Research supported by JSPS KAKENHI Grat Number Research partially supported by FCT - Fudação para a Ciêcia e a Tecologia (Portuguese Foudatio for Sciece ad Techology), through the projects PEst-OE/MAT/UI0297/2014 (Cetre for Mathematics ad its Applicatios) ad PTDC/MAT/113207/

2 Keywords: Edge-colourig; Gallai colourig; idepedece umber; graph partitioig; graph coverig; hypergraph 1 Itroductio I this survey article, we refer to the books by Bollobás [13] ad Berge [10] for ay udefied terms i graph theory ad hypergraph theory. For a graph G (resp. hypergraph H), the vertex set ad edge set are deoted by V (G) ad E(G) (resp. V (H) ad E(H)). Uless otherwise stated, all graphs ad hypergraphs are fiite, udirected, ad without multiple edges or loops. For a hypergraph H, a edge cosists of a subset of (uordered) vertices of H, ad whe we wish to emphasize that a edge has t vertices, we may call such a edge a t-edge. A o-empty t-uiform hypergraph is trivial if it cosists of fewer tha t isolated vertices, otherwise it is o-trivial. I particular, a trivial graph has a sigle vertex. The complete graph (or clique) o vertices is deoted by K, the cycle o vertices (or of legth ) is deoted by C, the path of legth l is deoted by P l, ad the complete bipartite graph with class-sizes m ad is deoted by K m,. The t-uiform complete hypergraph o vertices is deoted by K. t For a vertex v ad disjoit vertex subsets X, Y i a graph G, the degree of v ad the miimum degree of G are deoted by deg(v) ad δ(g), ad the bipartite subgraph iduced by X ad Y is deoted by (X, Y ). A idepedet set i a graph G (resp. hypergraph H) is a subset of vertices that does ot cotai a edge of G (resp. H). The idepedece umber of G, deoted by α(g), is the maximum cardiality of a idepedet set of G, ad α(h) is similarly defied for H. For a iteger r 1, a r-edge-colourig of a graph G, or simply a r-colourig, is a fuctio φ : E(G) {1, 2,..., r}. A similar defiitio holds for hypergraphs with H i place of G. Iformally, a r-colourig of G (resp. H) is a assigmet where every edge of G (resp. H) is give oe of r possible colours. Whe we do ot wish to emphasize the umber of colours ivolved, we may simply call such a assigmet a edge-colourig. I the case of a 2-colourig of a graph G, we ofte assume that the two colours are red ad blue. We say that the red subgraph is the subgraph R of G with V (R) = V (G) ad cosistig of the red edges, with a similar defiitio for the blue subgraph B with the blue edges. Fially, we will ofte deal with partitios of vertex sets of graphs ad hypergraphs, with all parts as equal as possible. For a set S with elemets, we call a partitio of S ito S 1,..., S p a ear-equal partitio if we have S i S j 1 for every 1 i, j p. Note that such a partitio is uique for fixed ad p, ad for every 1 i p, we have S i = p or Si = p. The subject of edge-colourigs of graphs has bee a well-studied topic of graph theory over the last several decades. A ladmark result is arguably Ramsey s theorem, published i 1930, for which the simplest versio states that: Give a iteger k 2, wheever the edges of a sufficietly large complete graph are coloured with red ad blue, there is a moochromatic copy of the complete graph K k. Sice the, a ear-edless amout of related research has bee carried out, leadig to the foudatio of several braches i the subject. For example, i (geeralised) graph Ramsey theory, oe is iterested i fidig moochromatic structures i edge-coloured graphs, ad a cetral problem is to determie the Ramsey umber R r (H), the miimum iteger such that for ay r-colourig of the complete graph K, there exists a moochromatic copy of the graph H. O the other had, the area of ati-ramsey theory, iitiated by Erdős, Simoovits ad Sós [28] (1973), deals with a similar cocept, where oe is iterested i raibow coloured 2

3 subgraphs (i.e. all edges of the subgraph have distict colours) i edge-coloured graphs. A fuctio of particular iterest is the ati-ramsey umber AR(, H), the maximum iteger r for which there exists a r-colourig of the complete graph K, such that there is o raibow copy of the graph H. Aother well-studied area is Ramsey-Turá theory, which is a area that features the coectio betwee Ramsey ad Turá type problems. Here, a fuctio of iterest is the Ramsey-Turá umber RT r (, H, k), which is the maximum umber of edges that a graph G o vertices ca have such that, G has o idepedet set of k vertices, ad there exists a r-colourig of G with o moochromatic copy of the graph H. This area probably emerged i the 1960s whe Adrásfai [4, 5] aswered some questios of Erdős that cocer the fuctio RT r (, H, k). Our aim i this survey is to cosider a class of problems which belog to the graph Ramsey theory area. We are maily iterested i two questios. Suppose that a graph is give a edge-colourig we sometimes call such a graph a host graph, ad usually thik of it as a rather large graph. The first questio is: O how may vertices ca we fid a moochromatic subgraph of a certai type, such as a coected subgraph, or a cycle, or some type of tree? The secod questio is: How may such moochromatic subgraphs do we eed so that their vertex sets form either a partitio or a coverig of the vertices of the host graph? We shall review kow results ad cojectures related to these two questios i Sectios 2 ad 3. I Sectio 4, we cosider the two questios i hypergraph settigs. Some of the work date as far back as the 1960s, ad the research had bee particularly itese sice aroud the 1990s. We will, i particular, preset may examples of edge-colourigs of graphs ad hypergraphs, where each edge-colourig shows either the sharpess of a result, or the support of a possible cojecture. Our results ad cojectures here will have some overlaps with recet surveys of Kao ad Li [67] (2008), Fujita, Magat ad Ozeki [39] (2010), ad Gyárfás [47] (2011). We will attempt to miimise these overlaps, ad also to emphasize more recet developmets. 2 Moochromatic Structures i Graphs I this sectio, we cosider the problem of fidig moochromatic subgraphs i edge-coloured graphs. A first result i this directio is the followig observatio, made a log time ago by Erdős ad Rado: A graph is either coected, or its complemet is coected. I other words, for every 2-colourig of the edges of a complete graph, there exists a moochromatic spaig coected subgraph (or equivaletly, a moochromatic spaig tree). A substatial geeralisatio of this observatio is to ask for the existece of a large moochromatic subgraph of a certai type i a edge-coloured graph. Here, we preset may kow results ad problems related to this questio. I may cases, the edge-coloured host graph is a complete graph. 2.1 Coected ad k-coected subgraphs To exted the observatio of Erdős ad Rado, oe way is to ask what happes whe r 2 colours are used to colour the complete graph K. I this directio, Gerecsér ad Gyárfás [43] (1967) asked for the order of a moochromatic coected subgraph that oe ca always fid. Gyárfás ad Füredi idepedetly proved the followig result (see also Liu et al. [77] for a short proof). 3

4 Theorem (Gyárfás [44]; Füredi [40]). Let r 2. For every r-colourig of K, there exists a moochromatic coected subgraph o at least r 1 vertices. This boud is sharp if r 1 is a prime power. To see the sharpess, we cosider the followig well-kow costructio of a r-colourig o K, usig affie plaes. This costructio will also be importat for may more problems that we will ecouter throughout the etire survey. Costructio Let r 1 be a prime power, ad cosider the fiite affie plae AG(2, r 1) over the field F r 1 (see the appedix i Sectio 5). Let p 1,..., p (r 1) 2 be the poits ad P 1,..., P r be the parallel classes of lies of AG(2, r 1). Now, take a ear-equal partitio of the vertex set of K ito (r 1) 2 classes V 1,..., V (r 1) 2. We defie the r-colourig ψ o K as follows. If u V i ad v V j are two vertices of K ad 1 i j (r 1) 2, the let ψ(uv) = l if ad oly if p i ad p j lie o the same lie i the class P l (where 1 l r). Colour the edges iside the classes V 1,..., V (r 1) 2 arbitrarily. The i the r-colourig ψ of K, every moochromatic coected subgraph has at most (r 1) (r 1) < 2 r 1 + r vertices. Also, a very oteworthy feature about the r-colourig ψ is that, it ca be obtaied by substitutig edge-coloured cliques (sometimes called blocks) for the vertices of a edge-coloured graph i this case a r-colourig of K (r 1) 2, with all edges betwee a pair of cliques retaiig the colour of the correspodig two substituted vertices. As we shall see, this colourig by substitutio techique will be importat i may more costructios of edge-colourigs. To exted Theorem 2.1.1, we may ask for the existece of a large moochromatic subgraph with high vertex coectivity i a edge-coloured complete graph. Recall that a graph G is k- coected if V (G) > k, ad for ay set C V (G) with C < k, the graph G C is coected. The followig questio was asked by Bollobás. Questio (Bollobás [14]). Let 1 s < r ad k 1. Wheever we have a r-colourig of the edges of K, o how may vertices ca we fid a k-coected subgraph, usig at most s colours? We shall focus o Bollobás questio oly for the case whe the desired subgraph is moochromatic, i.e. s = 1. For results cocerig s 2, we refer the reader to Liu et al. [76]. I order to state the results, we make the followig defiitio. Let m(, r, k) deote the maximum iteger m such that, every r-colourig of K cotais a moochromatic k-coected subgraph o at least m vertices. The, our task is to determie the fuctio m(, r, k). If 2r(k 1), the we have the followig costructio of Matula [82] (1983). Costructio Let 2r(k 1). We cosider the stadard decompositio of K 2r ito r edge-disjoit zig-zag rotatioal Hamilto paths (see for example [13], Ch. I, Theorem 11). That is, let v 1,..., v 2r be the vertices of K 2r, ad cosider the Hamilto path Q 1 = v 1 v 2r v 2 v 2r 1 v r v r+1. For 1 < l r, let Q l be the Hamilto path obtaied from Q 1 by addig l 1 to the idices of the vertices of Q 1, modulo 2r. The, Q 1,..., Q r is a decompositio of K 2r ito edge-disjoit Hamilto paths, ad moreover, every vertex of K 2r is the ed-vertex of exactly oe Hamilto path. We colour the Hamilto path Q l with colour l for all 1 l r. Now, we obtai a r-colourig of K as follows. Partitio the vertices of K ito 4

5 V 1,..., V 2r such that V i k 1 for all 1 i 2r. For 1 i j 2r, we give all edges of (V i, V j ) the colour of v i v j, ad all edges iside V i the colour of the Hamilto path i K 2r with v i as a ed-vertex (i.e. for 1 l r, all edges iside V l ad V r+l are give colour l). We see that i this r-colourig of K, there is o moochromatic k-coected subgraph at all, ad hece m(, r, k) = 0 for 2r(k 1). I the case r = 2, Bollobás ad Gyárfás [15] (2003) gave the followig costructio of a 2-colourig of K for > 4(k 1). Costructio Let > 4(k 1). We defie a 2-colourig o K as follows. Let V 1, V 2, V 3, V 4 be four disjoit sets of vertices of K, each with size k 1, ad let V 5 be the remaiig vertices (ote that V 5 ). We colour all edges of (V i, V j ) red if {i, j} {{1, 2}, {1, 3}, {2, 4}, {1, 5}, {2, 5}}, ad blue otherwise. The edges iside the classes V i are arbitrarily coloured. We see that i Costructios ad 2.1.5, the edge-colourigs of K are obtaied by the colourig by substitutio techique. I the case of the former, we have substituted moochromatic cliques for the vertices of a r-coloured K 2r. I the latter, the 2-colourig of K is obtaied by substitutig arbitrarily 2-coloured cliques for the vertices of K 5, where the K 5 is give a 2-colourig with both colour classes formig a bull (i.e. the graph with two pedat edges attached to two vertices of a triagle). I the 2-colourig of K i Costructio 2.1.5, it is easy to check that the moochromatic k-coected subgraph with maximum order has 2k+2 vertices. That is, we have m(, 2, k) 2k + 2 if > 4(k 1). Ispired by Costructio for r = 2 ad Costructio 2.1.5, Bollobás ad Gyárfás made the followig cojecture. Cojecture (Bollobás, Gyárfás [15]). For > 4(k 1), we have m(, 2, k) = 2k + 2. The two costructios imply that if Cojecture is true, the the boud > 4(k 1) is the best possible. May partial results to the cojecture are kow. The cojecture has bee verified by Bollobás ad Gyárfás for k = 2 [15]; ad by Liu et al. for k = 3 [75], ad for 13k 15 [77]. The best kow partial result is by Fujita ad Magat. Theorem (Fujita, Magat [36]). For > 6.5(k 1), we have m(, 2, k) = 2k + 2. Whe r 3 colours are used, Liu et al. also studied the fuctio m(, r, k). Theorem (Liu, Morris, Price [77]). Let r 3. (a) m(, r, k) r 1 11(k2 k)r. (b) If r 1 is a prime power, the m(, r, k) < k+1 r 1 + r. I particular, if r ad k are fixed ad r 1 is a prime power, the m(, r, k) = To obtai part (b), we ca easily modify Costructio 2.1.2, as follows. r 1 + O(1). 5

6 Costructio For r 3 ad > 2r(k 1), we defie the r-colourig ψ o K as follows. Take disjoit sets of vertices U 1,..., U r of K, each with k 1 vertices. Let W be the remaiig vertices (ote that W ), ad give the complete subgraph o W the r-colourig ψ as described i Costructio Now for 1 i j r, u U i, v U j ad w W, let ψ (uv) = mi{i, j} ad ψ (uw) = i. The edges iside the classes U i are arbitrarily coloured. It is easy to see that i the r-colourig ψ of K, the moochromatic k-coected subgraph with maximum order has at most (r 1) r(k 1) (r 1) + k 1 < k+1 2 r 1 + r vertices. Hece, m(, r, k) < k+1 r 1 +r for > 2r(k 1). Moreover, Costructio implies that m(, r, k) = 0 for 2r(k 1). I view of these two costructios, Liu et al. also made the followig cojecture. Cojecture (Liu, Morris, Price [77]). Let r 3 ad > 2r(k 1). The m(, r, k) k + 1. r 1 The cojecture has bee verified by Liu et al. for r = 3 ad large. Theorem (Liu, Morris, Price [77]). For 480k, we have k m(, 3, k) k Moreover, equality holds i the lower boud if ad oly if + k 1 (mod 4). We also see that Theorem 2.1.8(a) gives a good boud oly whe k = o( ). Liu ad Perso used Szemerédi s regularity lemma to obtai the followig improvemet. Theorem (Liu, Perso [78]). Let r 3 be fixed, ad k = o(). The, we have m(, r, k) o(). Equality holds if r 1 is a prime power. r 1 Fially, we metio a slightly differet versio of Questio 2.1.3, for s = 1. Let m (r, k) be the smallest iteger such that, ay r-colourig of K has a moochromatic k-coected subgraph. The problem of the determiatio of the fuctio m (r, k) was proposed by Matula. Note that i this problem, oe does ot worry about the order of a moochromatic k-coected subgraph i r-coloured complete graphs, but oly that such a subgraph exists. Matula proved the followig result. Theorem (Matula [82]). Let k 2. ( (a) 4(k 1) + 1 m (2, k) < ) (k 1) + 1. (b) For r 2, we have 2r(k 1) + 1 m (r, k) < 10 3 r(k 1) + 1. The lower bouds follow from Costructio Matula also made the followig cojecture. It is a slightly weaker assertio tha the combiatio of Cojectures ad Cojecture (Matula [82]). For r, k 2, we have m (r, k) = 2r(k 1)

7 Matula remarked that by usig some tedious argumets, Cojecture holds for k = 2, 3, 4, 5, ad that the upper boud of Theorem (a) ca be improved to somewhat below 4.7(k 1) + 1. Next, we cosider the aalogous problem whe the host graph is a complete bipartite graph. We may ask for the order of a moochromatic coected subgraph that we ca always fid i ay r-colourig of a complete bipartite graph K m,. For this, Gyárfás proved the followig result. Lemma (Gyárfás [44]). Let r 1. For every r-colourig of the complete bipartite graph K m,, there exists a moochromatic coected subgraph o at least m+ r vertices. Lemma easily implies the first part of Theorem Ideed, let K be give a r-colourig (r 2). If o colour spas a coected subgraph o vertices, the there is a colour that spas a coected compoet o vertex set X with X <. By lettig Y be the remaiig vertices, we may apply Lemma o the (r 1)-coloured complete bipartite subgraph (X, Y ), to obtai a moochromatic coected subgraph o at least r 1 vertices. To see the sharpess of Lemma , we may cosider the followig example. Costructio Take ear-equal partitios of each class of K m, ito r sets, say U 1,..., U r ad V 1,..., V r. For 1 i, j r, colour all edges of (U i, V j ) with colour i j (mod r). The i this r-colourig of K m,, the moochromatic coected subgraph with maximum order has at most m+ r + 2 vertices, sice we have U i m r + 1 ad V i r + 1 for all 1 i r. Ispired by Lemma , Liu et al. [77] made the cojecture that the same result holds whe we wat to fid a moochromatic k-coected subgraph, provided that the classes of K m, are large: For r 3 ad m, rk, ay r-colourig of K m, cotais a moochromatic k-coected subgraph o at least m+ r vertices. Note that the boud of m+ r does ot deped o k. Lemma implies that the cojecture holds for k = 1. The sharpess of the cojecture ca agai be see by Costructio ote that the coditio m, rk implies that, the moochromatic k-coected subgraph with maximum order i the r-colourig has at most m+ r + 2 vertices. However, we see that the case r = 2 is somehow omitted. Ideed, we preset the followig example of a r-colourig of K m,, which is a couter-example to Liu et al. s cojecture for small. Costructio Let r, k 2, 2r(k 1) ad m 6(r + 1). Partitio the m-class of K m, ito 2r 1 ear-equal sets, say U 1,..., U 2r 1, ad the -class ito 2r ear-equal sets, say V 1,..., V 2r. A result of Laskar ad Auerbach [72] (1976) implies that the complete bipartite graph K 2r 1,2r ca be decomposed ito r edge-disjoit Hamilto paths, where every vertex of the 2r-class is a ed-vertex of exactly oe Hamilto path. Colour the Hamilto paths with r distict colours, ad let {u 1,..., u 2r 1 } ad {v 1,..., v 2r } be the classes of the K 2r 1,2r. Now, for 1 i 2r 1 ad 1 j 2r, we give all edges of (U i, V j ) the colour of the edge u i v j. The, 2r(k 1) implies V j k 1 for all 1 j 2r. Hece i this r-colourig of K m,, the moochromatic k-coected subgraph of maximum order has at most m 2r r < m+ r vertices (the last iequality follows from m 6(r + 1)). I light of this costructio, we revise the cojecture of Liu et al., as follows. 7

8 Cojecture (Refiemet of Liu, Morris, Price [77]). Let r 2 ad m > 2r(k 1). The, for ay r-colourig of the complete bipartite graph K m,, there exists a moochromatic k-coected subgraph o at least m+ r vertices. By usig a bipartite versio of Szemerédi s regularity lemma, Liu ad Perso, i respose to Liu et al. s origial cojecture, obtaied the followig partial result. Theorem (Liu, Perso [78]). Let r 2 be fixed, m k, ad k = o(). The, for every r-colourig of the compete bipartite graph K m,, there exists a moochromatic k-coected subgraph o at least m+ r o() vertices. 2.2 Cycles ad regular subgraphs Give a edge-coloured graph, we may ask for the existece of a log moochromatic cycle. Recall that the circumferece of a graph G, deoted by c(g), is the legth of a logest cycle i G. Faudree et al. proved the followig result. Theorem (Faudree, Lesiak, Schiermeyer [29]). Let G be a graph of order 6, ad let G be the complemet of G. The max{c(g), c(g)} 2 3, ad this boud is sharp. The sharpess ca be easily see by takig G to be the graph cosistig of 3 isolated vertices ad a clique o the remaiig 2 3 vertices. Note that Theorem is equivalet to sayig that, i ay 2-colourig of K ( 6), there exists a moochromatic cycle with legth at least 2 3. Whe r 3 colours are used to colour K, we may agai cosider the r-colourig ψ o K, as described i Costructio The, if r 1 is a prime power, the logest moochromatic cycle has legth at most r 1 + r. Ispired by the costructio, Faudree et al. also made the followig cojecture. Cojecture (Faudree, Lesiak, Schiermeyer [29]). For r 3, let K be give a r-colourig. For 1 i r, let G i be the graph o vertices iduced by colour i. The max{c(g 1 ), c(g 2 ),..., c(g r )} r 1. Let f(, r) deote the maximum iteger l such that every r-colourig of K cotais a moochromatic cycle of legth at least l. The f(, r) < r 1 + r if r 1 is a prime power, while Cojecture claims that f(, r) r 1 wheever r 3. Fujita [31] (2011) observed that the cojecture does ot hold for small, by cosiderig the decompositio of K 2r ito r edge-disjoit zig-zag rotatioal Hamilto paths as described i Costructio 2.1.4, ad givig each Hamilto path a distict colour. Clearly, this r-colourig of K 2r implies f(, r) = 0 for 2r, sice by deletig 2r vertices from K 2r, there is o moochromatic cycle at all i the resultig r-colourig of K. Cojecture remais ope for sufficietly large certaily, we eed > 2r to hold. I the case whe is liear i r, Fujita et al. proved the followig result. Theorem (Fujita, Lesiak, Tóth [35]). (a) For r 3, we have f(2r + 2, r) = 3. For r = 1, 2, we have f(2r + 2, r) = 4. 8

9 (b) For ay pair of itegers s, c 2, there exists r 0 = r 0 (s, c) such that f(sr + c, r) = s + 1 for all r r 0. I particular, part (b) implies that Cojecture holds for = sr + c with r sufficietly large. By usig a result of Erdős ad Gallai [26] (1959), which is a Turá type result for the cycle, Fujita also obtaied the followig slightly weaker versio of the cojecture. Theorem (Fujita [31]). For r 1, we have f(, r) r. As a geeralisatio, we may ask for the largest order of a moochromatic coected d-regular subgraph, where d 2. Thus, the case d = 2 correspods to cycles. Sárközy et al. proved the followig result. Theorem (Sárközy, Selkow, Sog [91]). For every ε > 0 ad itegers r, d 2, there exists 0 = 0 (ε, r, d) such that the followig holds. For all 0 ad ay r-colourig of K, there exists a moochromatic coected d-regular subgraph o at least (1 ε) r vertices. We see that Theorem ca be see as a extesio of Theorem The slightly surprisig fact about this extesio is that the value of d plays a fairly mior role, i the sese that the order of a moochromatic coected d-regular subgraph that we ca fid is almost as large as that of a moochromatic cycle, i.e. approximately r, ad idepedet of d. Whe the edge-coloured host graph is ot complete, Li et al. proposed the followig problem, where there is a coditio o the miimum degree of the host graph. Problem (Li, Nikiforov, Schelp [74]). Let 0 < c < 1 ad let be a sufficietly large iteger. If G is a graph of order with δ(g) > c, ad G is give a 2-colourig with red ad blue subgraphs R ad B, what is the miimum possible value of max{c(r), c(b)}? As opposed to just fidig a sigle moochromatic cycle of a specified legth, some authors have cosidered the problem of fidig all cycle legths i a specified iterval. I this directio, Li et al. cojectured the followig pacyclic type result for moochromatic cycles i 2-coloured graphs. Cojecture (Li, Nikiforov, Schelp [74]). Let 4 ad G be a graph of order with δ(g) > 3 4. If G is give a 2-colourig with red ad blue subgraphs R ad B, the for each iteger l [ 4, ] 2, either Cl R or C l B. They observed the followig example, which shows that if this cojecture is true, the it is tight: Let = 4p. Colour the edges of the complete bipartite graph K 2p,2p red, ad add a blue K p,p i each of its classes. The we have a 2-coloured graph G with δ(g) = 3 4, but clearly the colourig produces o moochromatic odd cycle. I the same paper, Li et al. proved the followig partial result. Theorem (Li, Nikiforov, Schelp [74]). Let ε > 0 ad G be a graph of sufficietly large order with δ(g) > 3 4. If G is give a 2-colourig with red ad blue subgraphs R ad B, the for each iteger l [ 4, ( 1 8 ε) ], either C l R or C l B. 9

10 Beevides et al. offered the followig slightly stroger solutio but for large. Here we eed a defiitio. Let = 4p ad G be a graph isomorphic to K p,p,p,p. The G is said to be 2-bipartite 2-edge-coloured if the edges of G are 2-coloured so that the graph iduced by each of the colours is bipartite. Such a 2-bipartite 2-edge-coloured graph has miimum degree 3 4 but cotais o moochromatic odd cycles. Theorem (Beevides, Luczak, Scott, Skoka, White [9]). There exists a positive iteger 0 with the followig property. Let G be a graph of order 0 with δ(g) 3 4. Suppose that G is give a 2-colourig with red ad blue subgraphs R ad B. The either C l R or C l B for all l [ 4, ] 2, or = 4p, G = Kp,p,p,p ad the colourig is a 2-bipartite 2-edge-colourig. 2.3 Subgraphs with bouded diameter We cosider the problem of fidig moochromatic subgraphs with bouded diameter i edgecoloured complete graphs. For r, D 2, defie g(, r, D) to be the maximum iteger m such that, for every r-colourig of K, there is a moochromatic subgraph with diameter at most D o at least m vertices. We remark that we eglect the case D = 1, sice this is equivalet to the problem of the determiatio of the classical Ramsey umber R r (K m ). The problem of the determiatio of the fuctio g(, r, D) was proposed by Erdős (for D = 2), ad Mubayi (for D 3). Problem (Erdős [24]; Mubayi [83]). For r, D 2, determie the fuctio g(, r, D). By simply cosiderig the largest moochromatic star at ay vertex of a r-coloured K, Erdős oticed that g(, r, 2) 1 r + 1, ad asked if this boud is optimal. Fowler the proved that the aswer is yes, if ad oly if r 3. He determied g(, 2, 2) exactly, ad g(, r, 2) sharply for r 3. Theorem (Fowler [30]). (a) g(, 2, 2) = 3 4. (b) For fixed r 3, we have g(, r, 2) = r + O(1). To see that g(, 2, 2) 3 4, we may cosider the followig 2-colourig of K. Take a earequal partitio of the vertices of K ito sets V 1, V 2, V 3, V 4. Colour all edges of (V i, V i+1 ) i red for i = 1, 2, 3, all other edges betwee the classes i blue, ad the edges iside the classes arbitrarily. The, the subgraph with diameter at most 2 ad of maximum order has 3 4 vertices. The proofs of g(, 2, 2) 3 4 ad g(, r, 2) r + O(1) for r 3 by Fowler are more complicated. Now, we cosider D 3. For the case r = 2, the followig result is folkloristic, ad a early citatio ca be foud i Bialostocki et al. [12]. Theorem We have g(, 2, D) = for all D 3. That is, i every 2-colourig of K, there exists a moochromatic spaig subgraph with diameter at most D. For r, D 3, we agai have the upper boud of g(, r, D) < r 1 + r if r 1 is a prime power, from Costructio Mubayi maaged to prove a lower boud for g(, r, 3). 10

11 Theorem (Mubayi [83]). For r 2, we have g(, r, 3) > r r For r = 3 ad D 4, Mubayi also maaged to compute g(, 3, D) exactly. Theorem (Mubayi [83]). For D 4, we have g(, 3, D) = { if 2 (mod 4), 2 otherwise. Thus i Problem 2.3.1, the determiatio of the fuctio g(, r, D), i a sharpess sese for large, remais ope for r = D = 3, ad for r 4, D Subgraphs with large miimum degree I a edge-coloured graph, how large a moochromatic subgraph ca we fid, if the host graph ad the moochromatic subgraphs have costraits o their miimum degrees? Let h(, c, d, r) be the maximum iteger m such that, for every r-colourig of ay graph of order ad with miimum degree at least c, there exists a moochromatic subgraph with miimum degree at least d ad order at least m. Caro ad Yuster proposed the problem of the determiatio of h(, c, d, r). They proved the followig results. Theorem (Caro, Yuster [19]). (a) For all d 1 ad c > 4(d 1), h(, c, d, 2) c 4d + 4 3d(d 1) + 2(c 3d + 3) 4(c 3d + 3). (b) For all d 1 ad c 4(d 1), if is sufficietly large, the h(, c, d, 2) d 2 d + 1. I particular, h(, c, d, 2) is idepedet of. Theorem (Caro, Yuster [19]). For all d 1, r 2 ad c > 2r(d 1), there exists a costat C such that I particular, h(, c, d, 2) h(, c, d, r) c 4d+4 2(c 3d+3) + C. c 2r(d 1) r(c (r + 1)(d 1)) + C. We see that Theorems ad imply that if c is fixed, the h(, c, d, 2) is determied up to a costat additive term. The theorems also show that h(, c, d, 2) trasitios from a costat to a value liear i whe c icreases from 4d 4 to 4d 3. To see Theorem 2.4.1(b), it suffices to costruct a 2-coloured, 4(d 1)-regular graph o vertices (for sufficietly large), with o moochromatic subgraph havig miimum degree at least d ad o more tha d 2 d + 1 vertices. Caro ad Yuster gave the followig costructio. 11

12 Costructio Let c = 4(d 1) ad be sufficietly large. We first create a specific graph H o vertices. Let the vertices be v 1,..., v ad coect two vertices v i ad v j if ad oly if i j d 1. Hece, all the vertices v d,..., v d+1 have degree 2(d 1), ad the remaiig 2(d 1) vertices have smaller degree. We add the followig ( d 2) edges. For all 1 i j d 1, we add the edge v i v jd+1. For example, if d = 3 we add v 1 v 4, v 1 v 7 ad v 2 v 7. Note that these added edges are ideed ew edges. The resultig graph H has vertices ad (d 1) edges. Furthermore, all the vertices have degree 2(d 1), except for v jd+1 whose degree is 2(d 1) + j, ad v d+1+j whose degree is 2(d 1) j, for 1 j d 1. Now, ote that the vertices of excess degree, amely v d+1, v 2d+1,..., v d 2 d+1, form a idepedet set. Hece, for sufficietly large, there exists a 4(d 1)-regular graph with vertices, ad a 2-colourig of it, such that each moochromatic subgraph is isomorphic to H. I the secod copy, the vertex playig the role of v jd+1 plays the role of the vertex v d+1+j i the first copy, for 1 j d 1, ad vice versa. Observe that i the first copy of H, ay subgraph with miimum degree at least d may oly cotai the vertices v 1,..., v d 2 d+1, ad thus has order at most d 2 d + 1. This clearly implies Theorem 2.4.1(b). Theorem trivially holds for d = 1. For d 2, Caro ad Yuster provided the followig costructio. Costructio Let d, r 2 ad c > 2r(d 1). We costruct a r-coloured graph with = r(m + d) vertices ad miimum degree at least c, where m is a arbitrary elemet of some fixed ifiite arithmetic sequece whose differece ad first elemet are oly fuctios of c, d ad r. This r-coloured graph will have o moochromatic subgraph with miimum degree at least d ad more vertices tha the value stated i Theorem 2.4.2, ad this clearly suffices for the theorem. Let m be a sufficietly large positive iteger such that y = (r 1)(d 1) c (r + 1)(d 1) m is a iteger. Let A 1,..., A r, B 1,..., B r be pairwise disjoit sets of vertices, with A i = y ad B i = m + d y for 1 i r. I each B i, we place a graph with colour i, ad with miimum degree at least c (r 1)(d 1). I each A i, we place a (d 1)-degeerate graph with colour i, havig precisely d vertices of degree d 1 ad the rest are of degree 2(d 1). It is easy to show that such a graph exists. Deote by A i the y d vertices of A i with degree 2(d 1) ad A i = A i \ A i. Now for each j i, we place a bipartite graph with colour i, whose classes are A i ad A j B j. I this bipartite graph, the degree of all the vertices of A j B j is d 1, the degrees of all the vertices of A c (r+1)(d 1) i are at least r 1, ad the degrees of all vertices of A i are at least c r(d 1) r 1. This ca be doe for m sufficietly large sice c (r + 1)(d 1) c r(d 1) (y d) + d (d 1)(m + d). r 1 r 1 For m sufficietly large, we ca place all of these r(r 1) bipartite graphs such that their edge sets are pairwise disjoit (a immediate cosequece of Hall s Theorem). 12

13 I this costructio, the miimum degree of the graph is at least c. Furthermore, ay moochromatic subgraph with miimum degree at least d must be completely placed withi some B i. It follows that h(, c, d, r) m + d (r 1)(d 1) c (r + 1)(d 1) m = c 2r(d 1) r(c (r + 1)(d 1)) + C. Caro ad Yuster also maaged to determie h(, c, d, 2) wheever c is very close to, for sufficietly large. Theorem (Caro, Yuster [19]). Let c ad d be positive itegers. For sufficietly large, h(, c, d, 2) = 2d c + 3. To see the upper boud of Theorem for large, they gave the followig costructio. Costructio Let A, A, B be disjoit sets of vertices with A = B = 2d + c 3 ad A = 2(2d + c 3). Colour all edges withi A A red, ad all edges withi B ad betwee A ad B blue. Let A = {v 1,..., v 2d+c 3 } ad B = {u 1,..., u 2d+c 3 }. For 1 i 2d + c 3, colour the d 1 edges from u i to v i,..., v i+d 2 blue, ad the d 1 edges to v i+d 1,..., v i+2d 3 red, with the idices of the v j take modulo 2d + c 3. There are o other edges coectig A ad B. It is easy to verify that for sufficietly large, this graph is ( c)-regular, ad cotais o moochromatic subgraph with miimum degree at least d ad more tha 2d c + 3 vertices. 2.5 Specific trees We recall the observatio of Erdős ad Rado: Every 2-coloured complete graph cotais a moochromatic spaig tree. We may exted this observatio, by isistig that the moochromatic subgraph is a specific type of tree. There are may results i this directio. Bialostocki et al. proved the followig. Theorem (Bialostocki, Dierker, Voxma [12]). For every 2-colourig of K, there exists a mochromatic spaig tree with height at most 2. Theorem (Bialostocki, Dierker, Voxma [12]). For every 2-colourig of K, there exists a mochromatic spaig subdivided star, whose cetre has degree at most 1 2. The same authors had also cojectured that every 2-coloured K cotais a moochromatic spaig broom (A broom is a path with a star at oe ed). This cojecture was proved by Burr, but his proof was ufortuately upublished. A proof of Burr s result ca be foud i the survey of Gyárfás [47]. Theorem (Burr [16]). For every 2-colourig of K, there exists a mochromatic spaig broom. A double star is a graph obtaied by coectig the cetres of two vertex-disjoit stars with a edge. Mubayi, ad Liu et al. idepedetly proved the followig result, which is a extesio of Lemma

14 Lemma (Mubayi [83]; Liu, Morris, Price [77]). Let r 1. For every r-colourig of the complete bipartite graph K m,, there exists a moochromatic double star with at least m+ r vertices. The sharpess of Lemma ca agai be see by the r-colourig of K m, i Costructio Ispired by the lemma, Gyárfás ad Sárközy studied the aalogous problem of fidig a moochromatic double star i a r-coloured complete graph. They oticed the lemma implies that i ay r-colourig of K, either all colour classes iduce just oe compoet, or there is a moochromatic double star with at least r 1 vertices. They also asked the followig questio, which is the aalogue of Theorem for double stars. Questio (Gyárfás ad Sárközy [53]). Let r 3. For every r-colourig of K, is it true that there exists a moochromatic double star o at least r 1 vertices? They maaged to prove the followig weaker result. Theorem (Gyárfás ad Sárközy [53]). For r 2 ad every r-colourig of K, there exists a moochromatic double star o at least (r+1)+r 1 vertices. r 2 For the case r = 2, Theorem gives the existece of a moochromatic double star o at least vertices i ay 2-coloured K. By cosiderig radom graphs or Paley graphs, oe ca obtai a 2-colourig of K where the moochromatic double star of maximum order has O(1) vertices. The radom graphs costructio was i fact show implicitly by Erdős, Faudree, Gyárfás ad Schelp [25] (1989). Hece i Questio 2.5.5, the costrait r 3 is ecessary. 2.6 Gallai colourigs ad extesios I this subsectio, we shall cosider the task of fidig moochromatic subgraphs i edgecoloured complete graphs by puttig a restrictio o the edge-colourig. I [61], Gyárfás ad Simoyi defied a Gallai colourig to be a edge-colourig of a graph where o triagle is coloured with three distict colours. This model of colourig dates back to Gallai s paper [42] (1967), where he studied trasitively orietable graphs, ad the paper was subsequetly traslated ito Eglish by Maffray ad Preissma [81]. Gallai colourigs have sice bee studied (directly or idirectly) by may authors. Notably, Camero, Edmods ad Lovász [17, 18, 79] ecoutered these colourigs, whe they exteded the perfect graph theorem. Also, Körer, Simoyi ad Tuza [70, 71] called such a edge-colourig a Gallai partitio, ad they foud the colourigs to be relevat i a iformatio theoretic fuctio called the graph etropy. Fially, Gyárfás et al. [58] itroduced the Gallai-Ramsey umber, which is the aalogue of the classical Ramsey umber, but restricted to Gallai colourigs. Gallai colourigs geeralise 2-colourigs, ad these two types of edge-colourigs are very closely related. Ideed, a flagship result is the followig decompositio theorem, which Gallai [42], ad Camero ad Edmods [17] used implicitly, ad was properly restated by Gyárfás ad Simoyi [61]. Theorem (Gallai [42]). Ay Gallai colourig o a complete graph ca be obtaied by substitutig complete graphs with Gallai colourigs for the vertices of a 2-coloured complete graph o at least two vertices. 14

15 That is, give a complete graph K with a Gallai colourig, the followig is true. There exists a partitio of the vertices of K ito sets V 1,..., V p (for some p 2) such that, the edges withi every V i form a Gallai colourig, ad for every 1 i j p, all edges of (V i, V j ) have the same colour ad ca be oe of oly two possible colours. Such a decompositio is called a Gallai decompositio of K, ad the Gallai coloured complete graphs o V 1,..., V p are the blocks of the Gallai decompositio. Also, the 2-coloured complete graph o p vertices, say v 1,..., v p, with v i v j give the colour of (V i, V j ) for all 1 i j p, is the base graph of the decompositio. The followig result also appeared i [42]. Gyárfás ad Simoyi stated the result explicitly i [61], ad oted that Theorem follows from it. Theorem (Gallai [42]). Every Gallai colourig with at least three colours o a complete graph K has a colour which iduces a discoected subgraph o vertices. We see that Theorem illustrates the close coectio betwee Gallai colourigs ad 2-colourigs. The theorem is a importat tool which ca be used to show that, certai results which hold for 2-colourigs also hold for Gallai colourigs. For istace, recall the observatio of Erdős ad Rado: Every 2-coloured complete graph has a moochromatic spaig tree. Now, give a Gallai colourig o a complete graph K, we may apply Theorem to obtai a Gallai decompositio for K, ad the the observatio o the base graph, to obtai a moochromatic spaig tree for K. Thus, Erdős ad Rado s observatio exteds to: Every Gallai coloured complete graph has a moochromatic spaig tree. It turs out that some of the other results that we have already see ca also be exteded, icludig Theorems 2.5.1, 2.5.3, 2.3.2(a), 2.3.3, ad Theorem for r = 2. Theorem (Gyárfás, Simoyi [61]). For every Gallai colourig of K, there exists a mochromatic spaig tree with height at most 2. Theorem (Gyárfás, Simoyi [61]). For every Gallai colourig of K, there exists a mochromatic spaig broom. Theorem (Gyárfás, Sárközy, Sebő, Selkow [58]). For every Gallai colourig of K, there exists a moochromatic subgraph with diameter at most 2 o at least 3 4 vertices. This is best possible for every. Theorem (Gyárfás, Sárközy, Sebő, Selkow [58]). For every Gallai colourig of K, there exists a moochromatic spaig subgraph with diameter at most 3. Theorem (Gyárfás, Sárközy, Sebő, Selkow [58]). For every Gallai colourig of K, there exists a moochromatic double star with at least vertices. This is asymptotically best possible. O the other had, a example where such a extesio does ot apply is whe we wat to fid a large moochromatic star i a edge-coloured complete graph. Give a 2-colourig of K, there is a moochromatic star o at least = vertices, by cosiderig the larger moochromatic star at ay vertex. This boud is essetially best possible, sice i the 2-colourig of K cosistig of two red cliques of orders 2 ad 2, with the remaiig edges blue, the moochromatic star with maximum order has vertices. With a little more effort, it is ot hard to show that we ca always fid a moochromatic star o vertices if 3 (mod 4), ad vertices if 3 (mod 4), with each value the best possible. However, we have the followig result for Gallai colourigs. 15

16 Theorem (Gyárfás, Simoyi [61]). For every Gallai colourig of K, there exists a moochromatic star with at least 2 5 vertices. This boud is sharp. The sharpess i Theorem ca be see from the followig Gallai colourig of K. Partitio the vertices of K ito five ear-equal sets V 1,..., V 5. Colour all edges of (V i, V i+1 ) with colour 1, those of (V i, V i+2 ) with colour 2, ad those iside the classes V i with colour 3 (for all 1 i 5, with idices take modulo 5). The, the moochromatic star of maximum order has vertices. To geeralise the cocept of Gallai colourigs, we may replace the role of the forbidde 3- coloured triagle. A edge-colourig of a graph F is raibow if the colours of the edges of F are distict. The, we may cosider edge-colourigs of complete graphs where a raibow coloured copy of a fixed graph F is forbidde. We shall call such a edge-colourig raibow F -free. I this directio, Fujita ad Magat cosidered graphs F that are close to a triagle. For s, t 0, let H s,t be the graph obtaied by takig a triagle, s sigle edges ad t copies of P 2 (the path of legth 2), ad idetifyig oe vertex of the triagle, say v, with oe ed-vertex of each sigle edge ad each P 2. The vertex v is the cetre of H s,t. Note that the s sigle edges are pedat edges of H s,t (recall that a pedat edge of a graph F is a edge that has a ed-vertex with degree 1 i F ). Fujita ad Magat cosidered aalogues of Theorem with raibow H s,0 -free colourigs. For s = 1, they proved the followig result, which shows that i a raibow H 1,0 -free colourig, oe caot hope to have a decompositio as strog as a Gallai decompositio. Theorem (Fujita, Magat [37]). For every raibow H 1,0 -free colourig of a complete graph K, oe of the followig holds. (i) V (K) ca be partitioed such that there are at most two colours o the edges betwee the parts. (ii) There are three (differet coloured) moochromatic spaig trees of K, ad moreover, there exists a partitio of V (K) with exactly three colours o edges betwee parts ad betwee each pair of parts, the edges have oly oe colour. For s 2, they also proved the followig decompositio type result for H s,0 -free colourigs. The decompositio is slightly weaker tha the decompositio whe s = 1 but it is still the best possible. Theorem (Fujita, Magat [37]). For s 2, i ay raibow H s,0 -free colourig of a complete graph K, there exists a partitio of V (K) such that betwee the parts, there are at most s + 2 colours. Furthermore, there exists a edge-colourig of a complete graph such that for every partitio of the vertices, there are s + 2 colours betwee the parts. Next, Fujita et al. cosidered extedig Theorem A graph F is said to have the discoectio property if there exists a iteger r 0 = r 0 (F ) such that the followig is true: For every complete graph K whose edges are coloured with at least r 0 colours ad without a raibow copy of F, there exists a colour which spas a discoected subgraph o vertices. Notice that r 0 (F ) E(F ), sice for every sufficietly large complete graph K ad every r E(F ) 1, there exists a r-colourig of K where every colour spas a coected subgraph, ad such a 16

17 r-colourig does ot cotai a raibow copy of F. If it is possible to take r 0 (F ) = E(F ), the F is said to have the Gallai property. Let DP ad GP deote, respectively, the family of graphs that have the discoectio property ad the Gallai property. Note that we have the followig. GP DP. If F is a subgraph of F ad F DP, the F DP. Combiig Theorems 2.6.2, ad , we have H s,0 GP for all s 0. Fujita et al. proved several results, which we summarise as follows. Theorem (Fujita, Gyárfás, Magat, Seress [34]). (a) Let P l deote the path of legth l. The P 2, P 3, P 4, P 5 GP. (b) Let C l deote the cycle of legth l. The C 2h GP for every h 1. (c) If F DP is coected ad bipartite, the F is either a tree, or a uicyclic graph, or two such compoets joied by a edge. (d) For ay F DP, there exists a edge e E(F ) such that F e is bipartite. (e) If F DP is coected, the F ca be obtaied from a tree by addig at most two edges. (f) If F is a uicyclic graph such that its cycle is a triagle, the F DP. Hece, ay forest belogs to DP. (g) H s,1 GP for all s 0. (h) H s,0 + GP for all s 0, where H+ s,0 is obtaied from H s,0 by addig a pedat edge to H s,0 at a vertex of the triagle, differet from the cetre. H1,0 2+ GP, where H2+ 1,0 is a triagle with three pedat edges added, oe at each vertex of the triagle. To see part (b), Fujita et al. preseted the followig costructio. Costructio Let A, B be disjoit sets with A = B = 2(r 1)q + 1. Let A = r 1 i=1 A i {a} ad B = r 1 i=1 B i {b}, where the sets are all disjoit ad A i = B i = 2q. The edge ab ad the edges withi A ad B are give colour r. For i = 1, 2,..., r 1, the edges betwee a ad B i ; b ad A i ; ad A i ad B i, are give colour i. Split each A i, B i ito two disjoit equal parts, A i = X i Y i, B i = U i W i (with q vertices i each). For ay i j {1, 2,..., r 1}, the edges betwee X i ad U j ; ad Y i ad W j, are give colour i; the edges betwee X i ad W j ; ad Y i ad U j are give colour j. This colours all edges of the complete graph o A B. By takig r = 2h, Fujita et al. showed that the r-colourig i Costructio is raibow C 2h -free, ad every colour iduces a coected subgraph. I light of their fidigs, Fujita et al. also stated the followig problems. The first is i respose to Theorem (a), the secod i respose to Theorem (a) ad Costructio , ad the fifth i respose to Theorem (g). 17

18 Problem (Fujita, Gyárfás, Magat, Seress [34]). (a) Are all paths i GP? (b) Is C 4 DP? (c) Are odd cycles i GP? Or i DP? (d) Do we have GP = DP? (e) Is H s,t GP for all s, t 0? Fially, Fujita ad Magat proved the followig result about the existece of moochromatic k-coected subgraphs i Gallai colourigs. Theorem (Fujita, Magat [38]). Let r 3 ad k 2. If (r + 11)(k 1) + 7k log k, the for every Gallai colourig of K usig r colours, there exists a moochromatic k-coected subgraph o at least r(k 1) vertices. Theorem ca be see as a extesio of Theorem It also illustrates the strikig differece that the Gallai colourig coditio imposes. If we cosider r ad k to be fixed, the Theorem gives, i a Gallai coloured K, the existece of a moochromatic k-coected subgraph o O(1) vertices, i.e. early all vertices of K. But Theorem 2.1.8(b) (hece Costructio 2.1.9) implies that we caot have more tha + O(1) vertices for such a subgraph i a r-coloured K, where r r is the largest iteger such that r 1 is a prime power. We ed this subsectio with the followig two results, where raibow coloured paths are avoided i edge-coloured complete graphs. Theorem (Thomaso, Wager [95]). Let r 4 ad k 1. If 2k, the every raibow P 4 -free r-colourig of K cotais a moochromatic k-coected subgraph o at least k + 1 vertices (where P 4 deotes the path of legth 4). Theorem (Fujita, Magat [38]). Let r max{ k 2 + 8, 15} ad k 1. If (r + 11)(k 1) + 7k log k + 2r + 3, the every raibow P 5 -free r-colourig of K cotais a moochromatic k-coected subgraph o at least 7k + 2 vertices (where P 5 deotes the path of legth 5). 2.7 Host graphs with give idepedece umber We give a brief review of some results of Gyárfás ad Sárközy, cocerig the existece of moochromatic coected subgraphs i edge-coloured graphs with give idepedece umber. Theorem (Gyárfás, Sárközy [54]). Let G be a graph o vertices ad with idepedece umber α(g) = α. The, for every 2-colourig of G, there exists a moochromatic coected subgraph o at least α vertices. This result is sharp. The sharpess i Theorem ca be easily see by takig G to cosist of α cliques of ear-equal orders, i.e. each clique has α or α vertices. Gyárfás ad Sárközy remarked that Theorem ca be exteded to r-colourigs, with α(r 1) i the role of α. They also cosidered usig Gallai colourigs, ad obtaied the followig result. 18 r 1

19 Theorem (Gyárfás, Sárközy [54]). Let G be a graph o vertices ad with idepedece umber α(g) = α. The, for every Gallai colourig of G, there exists a moochromatic coected subgraph o at least α 2 +α 1 vertices. They oted that the boud of i Theorem is ot far from the truth, ad provided α 2 +α 1 (c log α) Costructio below which shows that we caot have more tha for the maximum α 2 order of a moochromatic coected subgraph (for some costat c). Hece, the boud of α i Theorem does ot exted to Gallai colourigs. Costructio Cosider a triagle-free graph G with α(g ) = α ad with the maximum umber of vertices. That is, G has p = R(3, α + 1) 1 vertices, where R(3, α + 1) is the Ramsey umber of a triagle versus a K α+1 clique. A famous result of Kim [68] (1995) implies that p is α almost quadratic, its order of magitude is 2 log α. We give G a edge-colourig where all edges have distict colours. Now, we defie a edge-coloured graph G o vertices, by substitutig Gallai coloured cliques for the vertices of G, with the vertex sets of the cliques formig a earequal partitio of V (G). The, we have a Gallai colourig of G, ad α(g) = α. Moreover, the moochromatic coected subgraph of G with maximum order has at most 2 p = (c log α) vertices, where c is α 2 a costat comig from Kim s estimate of R(3, α + 1). Gyárfás ad Sárközy thus posed the followig problem. Problem (Gyárfás, Sárközy [54]). Determie the fuctio f(α), the largest value such that for every Gallai coloured graph G o vertices with idepedece umber α(g) = α, there exists a moochromatic coected subgraph o at least f(α) vertices. From Theorem ad Costructio 2.7.3, we have 1 c log α α 2 f(α) + α 1 α 2. For α = 2, Gyárfás ad Sárközy gave the followig costructio. Costructio Cosider the graph H 8 o eight vertices which is the complemet of the Wager graph, i.e. V (H 8 ) = {v 1,..., v 8 } ad E(H 8 ) = {v i v i±2, v i v i±3 : 1 i 8}, where idices are take modulo 8. Defie the edge-colourig o H 8 where for 1 i 8, the edges v i v i+2 ad v i v i 3 have colour i. As before, defie a edge-coloured graph G o vertices, by substitutig Gallai coloured cliques for the vertices of H 8, with the vertex sets of the cliques formig a ear-equal partitio of V (G). The, we have a Gallai colourig of G, ad α(g) = 2. Moreover, the moochromatic coected subgraph of G with maximum order has at most 3 8 vertices. Hece, we have the followig. Lemma (Gyárfás, Sárközy [54]). 1 5 f(2) 3 8. Fially, the followig result about the existece of a moochromatic double star i a Gallai coloured graph was also proved. Theorem (Gyárfás, Sárközy [54]). Let G be a graph o vertices ad with idepedece umber α(g) = α. The, for every Gallai colourig of G, there exists a moochromatic double star o at least α 2 +α 2/3 vertices. 19