Ramsey numbers for local colorings

Gyárfás, András; Lehel, Jenö; Schelp, R. H.; Tuza, Zs.

doi:10.1007/bf01788549

Cited by 56 publications

(49 citation statements)

References 8 publications

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“…We call an edge colouring simple, if V (C i ) ∩ V (C j ) = ∅ for all colours i, j that appear on an edge. In [15] it was shown that the number of colours in a simple 2-local colouring of K n is bounded by 3. In the next lemma we will see that for K n,n the number of colours in a simple 2-local colouring is bounded by 4.…”

Section: Bipartite Graphs With 2-local Colouringsmentioning

confidence: 99%

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Local colourings and monochromatic partitions in complete bipartite graphs

Lang

Stein

2015

Electronic Notes in Discrete Mathematics

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We show that for any 2-local colouring of the edges of the balanced complete bipartite graph Kn,n, its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite r-locally coloured graph with O(r 2 ) disjoint monochromatic cycles. We also determine the 2-local bipartite Ramsey number of a path almost exactly: Every 2-local colouring of the edges of Kn,n contains a monochromatic path on n vertices. MSC: 05C38, 05C55.

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Section: Bipartite Graphs With 2-local Colouringsmentioning

confidence: 99%

“…A colouring is rlocal if no vertex is adjacent to more than r edges of distinct colours. Local colorings have appeared mostly in the context of Ramsey theory [4,5,14,15,25,30,31,32].…”

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confidence: 99%

Local colourings and monochromatic partitions in complete bipartite graphs

Lang

Stein

2015

Electronic Notes in Discrete Mathematics

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“…Comparisons between r 2 (G) and r 2− oc (G) have been studied extensively in [18]. In particular the following theorem appears there.…”

Section: Rh Schelpmentioning

confidence: 99%

“…The concept of local colorings first appeared in a paper of Erdős and Sós [10] and was later studied extensively in [18]. It has also been more recently rediscovered by Galluccio, Simonovits, and Simonyi [17].…”

Section: Rh Schelpmentioning

confidence: 99%

Three edge-coloring conjectures

Schelp¹

2002

Discuss. Math. Graph Theory

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The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold. Keywords: edge-coloring, Ramsey number, vertex-distinguishing edge-coloring, strong chromatic index, balanced edge-coloring, local coloring, mean coloring. 2000 Mathematics Subject Classifications: 05C15, 05C55, 05C78.The purpose of this article is to give additional attention to three edgecoloring conjectures that have been considered over the last several years. Each conjecture together with closely related results appear in a separate section of the paper. For the reader's convenience each conjecture itself is set off in boldface type. A Conjecture on the Classical Ramsey NumberThe classical Ramsey number r k (G) is well known and is the smallest positive integer m such that any edge-coloring of K m by k colors contains a monchromatic copy of G. The conjecture to be posed allows a less restrictive edge-coloring of K m with the same consequence, a monochromatic copy of G in the colored K m . To be precise two special colorings need to be defined, the k-local coloring and the k-mean coloring.

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“…We also use the concept of local k-coloring introduced in Gyárfás et al [6]. An edge coloring of a graph (using any number of colors) is called a local k-coloring if the set of all edges incident with any given vertex are colored by at most k colors.…”

Section: Introductionmentioning

confidence: 99%

Finding a monochromatic subgraph or a rainbow path

Gyárfás

Lehel

Schelp

2006

Journal of Graph Theory

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For simple graphs G and H, let f(G, H) denote the least integer N such that every coloring of the edges of K N contains either a monochromatic copy of G or a rainbow copy of H. Here we investigate f(G, H) when H = P k . We show that even if the number of colors is unrestricted when defining f (G, H), the function f(G, P k ), for k = 4 and 5, equals the (k − 2)-coloring diagonal Ramsey number of G.

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Ramsey numbers for local colorings

Cited by 56 publications

References 8 publications

Local colourings and monochromatic partitions in complete bipartite graphs

Local colourings and monochromatic partitions in complete bipartite graphs

Three edge-coloring conjectures

Finding a monochromatic subgraph or a rainbow path

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