Large monochromatic components in multicolored bipartite graphs

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{m,n}$ there is a monochromatic connected component with at least ${m+n\over r}$ 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 $r$-coloring of the edges of an $(X,Y)$-bipartite graph with $|X|=m$, $|Y|=n$, $\delta(X,Y) > \left( 1 - \frac{1}{r+1}\right) n$ and $\delta(Y,X) > \left( 1 - \frac{1}{r+1}\right) m$, there exists a monochromatic component on at least $\frac{m+n}{r}$ vertices (as in the complete bipartite graph). If true, the minimum degree condition is sharp (in that both inequalities cannot be made weak when $m$ and $n$ are divisible by $r+1$). We prove the conjecture for $r=2$ and we prove a weaker bound for all $r\geq 3$. As a corollary, we obtain a result about the existence of monochromatic components with at least $\frac{n}{r-1}$ vertices in $r$-colored graphs with large minimum degree.