Rainbow triangles and cliques in edge-colored graphs

Ehard, Stefan; Mohr, Elena

doi:10.1016/j.ejc.2019.103037

Cited by 15 publications

(7 citation statements)

References 11 publications

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“…Moreover, as in the previous section, if one can show that for every possible clique-decomposition D of K n , (K n , D) admits a n-colouring, then the EFL Conjecture would be true. Although a proof of the conjecture for all sufficiently large values of n was recently announced [7], we still believe that such a problem deserves to be studied further and solved for the other instances as well, as this could give insights into related areas such as clique-decompositions and edge-colourings of graphs, which have been already studied such as in [3,9]. In this sense, we suggest the following problem which we think could be a possible way forward.…”

Section: Resultsmentioning

confidence: 89%

The Erdős--Faber--Lovász Conjecture revisited

Gauci,

Zerafa

2021

Preprint

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The Erdős-Faber-Lovász Conjecture, posed in 1972, states that if a graph G is the union of n cliques of order n (referred to as defining n-cliques) such that two cliques can share at most one vertex, then the vertices of G can be properly coloured using n colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining n-cliques. We here provide a quick and easy algorithm to colour the vertices of G in this case, and discuss connections with clique-decompositions and edge-colourings of graphs.

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Section: Resultsmentioning

confidence: 89%

The Erdős--Faber--Lovász Conjecture revisited

Gauci,

Zerafa

2021

Preprint

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“…2 − 1 but containing no rainbow triangles. Ehard et al [9] proved that if G is an edge-colored graph of order n with…”

Section: Introductionmentioning

confidence: 99%

Rainbow triangles in arc-colored tournaments

Zhang

Bai

et al. 2018

Preprint

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Let T n be an arc-colored tournament of order n. The maximum monochromatic indegree ∆ −mon (T n ) (resp. outdegree ∆ +mon (T n )) of T n is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of T n . The irregularity i(T n ) of T n is the maximum difference between the indegree and outdegree of a vertex of T n . A subdigraph H of an arc-colored digraph D is called rainbow if each pair of arcs in H have distinct colors. In this paper, we show that each vertex v in an arc-colored tournamentWe also give some maximum monochromatic degree conditions for T n to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. Finally, we present some examples showing that some of the conditions in our results are best possible.

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“…Finally, there are many natural variants or particular cases of this problem that have been studied and interesting open questions are pervasive. We conclude with a selection of these: forbidding rainbow triangles was considered in [1] and [11]; forbidding nonmonochromatic triangles was considered in [3]; a survey of a more general problem with weights was given in [9]; an inverted version of the problem was asked in [2]; finding multiple rainbow cliques was studied in [6].…”

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confidence: 99%

On a colored Turán problem of Diwan and Mubayi

Lamaison¹,

Müyesser²,

Tait³

2020

Preprint

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Suppose that R (red) and B (blue) are two graphs on the same vertex set of size n, and H is some graph with a red-blue coloring of its edges. How large can R and B be if R ∪ B does not contain a copy of H? Call the largest such integer mex(n, H). This problem was introduced by Diwan and Mubayi, who conjectured that (except for a few specific exceptions) when H is a complete graph on k + 1 vertices with any coloring of its edges mex(n, H) = ex(n, K k+1 ). This conjecture generalizes Turán's theorem.Diwan and Mubayi also asked for an analogue of Erdős-Stone-Simonovits theorem in this context. We prove the following asymptotic characterization of the extremal threshold in terms of the chromatic number χ(H) and the reduced maximum matching number M(H) of H.M(H) is, among the set of proper χ(H)-colorings of H, the largest set of disjoint pairs of color classes where each pair is connected by edges of just a single color. The result is also proved for more than 2 colors and is tight up to the implied constant factor.We also study mex(n, H) when H is a cycle with a red-blue coloring of its edges, and we show that mex(n, H) 1 2 n 2 , which is tight.

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Rainbow triangles and cliques in edge-colored graphs

Cited by 15 publications

References 11 publications

The Erdős--Faber--Lovász Conjecture revisited

The Erdős--Faber--Lovász Conjecture revisited

Rainbow triangles in arc-colored tournaments

On a colored Turán problem of Diwan and Mubayi

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