Ramsey number of paths and connected matchings in Ore-type host graphs

János Barát, András Gyárfás, Jeno Lehel, Gábor N. Sárközy

Research output: Contribution to journalArticle

Abstract

It is well-known (as a special case of the path-path Ramsey number) that in every 2-coloring of the edges of K3n-1, the complete graph on 3n-1 vertices, there is a monochromatic P2n, a path on 2n vertices. Schelp conjectured that this statement remains true if K3n-1 is replaced by any host graph on 3n-1 vertices with minimum degree at least 3(3n-1)4. Here we propose the following stronger conjecture, allowing host graphs with the corresponding Ore-type condition: If G is a graph on 3n-1 vertices such that for any two non-adjacent vertices u and v, dG(u)+dG(v)≥32(3n-1), then in any 2-coloring of the edges of G there is a monochromatic path on 2n vertices. Our main result proves the conjecture in a weaker form, replacing P2n by a connected matching of size n. Here a monochromatic, say red, matching in a 2-coloring of the edges of a graph is connected if its edges are all in the same connected component of the graph defined by the red edges. Applying the standard technique of converting connected matchings to paths with the Regularity Lemma, we use this result to get an asymptotic version of our conjecture for paths.

Original languageEnglish
Pages (from-to)1690-1698
Number of pages9
JournalDiscrete Mathematics
Volume339
Issue number6
DOIs
Publication statusPublished - Jun 6 2016

Keywords

  • Connected matchings
  • Ore-type graphs
  • Paths
  • Ramsey numbers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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