A Ramsey-type problem in directed and bipartite graphs

Gyárfás, András; Lehel, Jenö

doi:10.1007/bf02018597

Cited by 64 publications

(66 citation statements)

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“…The above argument extends easily to more colours, showing that every k-colouring of any tournament on n k + 1 vertices contains a monochromatic path of length n. This was observed by Chvátal [9] and Gyárfás and Lehel [17] who also obtained a similar result for paths of different lengths. Gyárfás and Lehel also observed that there is a simple grid construction that shows that the bound n k + 1 is tight.…”

Section: Introductionsupporting

confidence: 67%

“…We will use the following result of Gyárfás and Lehel [17] and Williamson [28], which finds the exact Ramsey number for two directed paths, in several places in our proof, although our methods would give an alternative argument for a linear bound on this Ramsey number. Note that this is a special case of Theorem 3.1 for two colours and directed paths.…”

Section: Preliminariesmentioning

confidence: 99%

“…Harary and Hell [19] introduced the notion of directed Ramsey numbers (for two colours) and determined its value for certain small graphs. Gyárfás and Lehel [17] and independently Williamson [28], deduce from a result of Raynaud [24] that ← → r ( − → P n , 2) = 2n − 3 for n ≥ 3. Bermond studied a related question for more colours; specifically, he considered the directed Ramsey number of a Hamiltonian graph (in one colour) vs. directed paths (of distinct lengths; in the remaining colours).…”

Section: Introductionmentioning

confidence: 97%

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Directed Ramsey number for trees

Bucić

Letzter

Sudakov

2017

Electronic Notes in Discrete Mathematics

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In this paper, we study Ramsey-type problems for directed graphs. We first consider the k-colour oriented Ramsey number of H, denoted by − → r (H, k), which is the least n for which every k-edgecoloured tournament on n vertices contains a monochromatic copy of H. We prove that − → r (T, k) ≤ c k |T | k for any oriented tree T . This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor.We also consider the k-colour directed Ramsey number ← → r (H, k) of H, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order n. Here we show that ← → r (T, k) ≤ c k |T | k−1 for any oriented tree T , which is again tight up to a constant factor, and it generalises a result by Williamson and by Gyárfás and Lehel who determined the 2-colour directed Ramsey number of directed paths.

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Section: Introductionsupporting

confidence: 67%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Directed Ramsey number for trees

Bucić

Letzter

Sudakov

2017

Electronic Notes in Discrete Mathematics

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“…Path-path Ramsey numbers for complete bipartite graphs were independently est~blished in [3] and [4]. The heart ofthe method in [4] can be stated as a vertex covering result (Theorem 4 below).…”

Section: Proof( Algorithm 2)mentioning

confidence: 99%

“…The heart ofthe method in [4] can be stated as a vertex covering result (Theorem 4 below). An exceptional coloring of a complete bipartite graph with vertex classes A and B is a coloring, where Theorem 4.…”

Section: Proof( Algorithm 2)mentioning

confidence: 99%

Vertex coverings by monochromatic paths and cycles

Gyárfás

1983

Journal of Graph Theory

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We survey some results on covering the vertices of 2-colored complete graphs by two paths or by two cycles of different color. We show the role of these results in determining path Ramsey numbers and in algorithms for finding long monochromatic paths or cycles in 2-colored complete graphs.A graph is 2-colored if each edge is colored either red or blue. We consider results on covering the vertices of 2-colored complete (undirected, directed, or bipartite) graphs by monochromatic paths or cycles. Algorithmic proofs of these results are given, and these results are applied to the calculation of generalized Ramsey numbers.A 2-colored path or cycle is called simple if it is monochromatic or it is the union of a red and a blue path. The following simple result was mentioned in a footnote in [ 2]. Proof (Algorithm 1). Assume that n ~ 2.-We define simple paths Pi,

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Calculating Ramsey Numbers by Partitioning Colored Graphs

Pokrovskiy

2016

Journal of Graph Theory

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Abstract:In this article we prove a new result about partitioning colored complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k ≥ 1, in every edge coloring of K n with the colors red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k + 1)-partite graph. When the coloring of K n is connected in red, we prove a stronger result-that it is possible to cover all the vertices with k red paths and a blue balanced complete (k + 2)-partite graph. Using these results we determine the Ramsey number of an n-vertex path, P n , versus a balanced complete tpartite graph on t m vertices, K t m , whenever m ≡ 1 (mod n − 1). We show that in this case R(P n , K t m ) = (t − 1)(n − 1) + t (m − 1) + 1, generalizing a result of Erdős who proved the m = 1 case of this result. We also determine the Ramsey number of a path P n versus the power of a path P t n . We show that R(P n , P t n ) = t (n − 1) + n t +1, solving a conjecture of Allen, Brightwell, and Skokan.

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A Ramsey-type problem in directed and bipartite graphs

Cited by 64 publications

References 1 publication

Directed Ramsey number for trees

Directed Ramsey number for trees

Vertex coverings by monochromatic paths and cycles

Calculating Ramsey Numbers by Partitioning Colored Graphs

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