Large monochromatic components in edge colored graphs with a minimum degree condition

A. Gyárfás, Gábor N. Sárközy

Research output: Contribution to journalArticle

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Abstract

It is well-known that in every k-coloring of the edges of the complete graph Kn there is a monochromatic connected component of order at least (formula presented)k-1. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For k = 2 the authors proved that δ(G) ≥(formula presented) ensures a monochromatic connected component with at least δ(G) + 1 vertices in every 2-coloring of the edges of a graph G with n vertices. This result is sharp, thus for k = 2 we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of k the situation is different, graphs of minimum degree (1 − ϵk)n can replace complete graphs and still there is a monochromatic connected component of order at least (formula presented), in fact (formula presented) suffices. Our second result is an improvement of this bound for k = 3. If the edges of G with δ(G) ≥ (formula presented) are 3-colored, then there is a monochromatic component of order at least n/2. We conjecture that this can be improved to 9 and for general k we (onjectu) the following: if k ≥ 3 and G is a graph of order n such that δ(G) ≥ (formula presented) n, then in any k-coloring of the edges of G there is a monochromatic connected component of order at least (formula presented).

Original languageEnglish
Article number#P3.54
JournalElectronic Journal of Combinatorics
Volume24
Issue number3
Publication statusPublished - Sep 8 2017

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Degree Condition
Edge-colored Graph
Minimum Degree
Coloring
Connected Components
Complete Graph
Colouring
Graph in graph theory
Color
Spanning Subgraph

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Theory and Mathematics

Cite this

Large monochromatic components in edge colored graphs with a minimum degree condition. / Gyárfás, A.; Sárközy, Gábor N.

In: Electronic Journal of Combinatorics, Vol. 24, No. 3, #P3.54, 08.09.2017.

Research output: Contribution to journalArticle

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