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

Gyárfás, András and Sárközy, Gábor (2017) Large monochromatic components in edge colored graphs with a minimum degree condition. ELECTRONIC JOURNAL OF COMBINATORICS, 24 (3). pp. 1-14. ISSN 1097-1440

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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). © 2017, Australian National University. All rights reserved.

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