Large monochromatic components in multicolored bipartite graphs

Abstract

AbstractIt is well known that in every ‐coloring of the edges of the complete bipartite graph there is a monochromatic connected component with at least vertices. In this paper we study an extension of this problem by replacing complete bipartite graphs by bipartite graphs of large minimum degree. We conjecture that in every ‐coloring of the edges of an ‐bipartite graph with , and , there exists a monochromatic component on at least vertices (as in the complete bipartite graph). If true, the minimum degree condition is sharp (in that both inequalities cannot be made weak when and are divisible by ). We prove the conjecture for r = 2 and we prove a weaker bound for all r ≥ 3. As a corollary, we obtain a result about the existence of monochromatic components with at least n/(r − 1) vertices in r‐colored graphs with large minimum degree.