Large monochromatic components in multicolored bipartite graphs

Abstract

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 false( m + n false) / 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 false| X false| = m , false| Y false| = n , δ ( X , Y ) > false( 1 − 1 / ( r + 1 ) false) n, and δ ( Y , X ) > false( 1 − 1 / false( r + 1 false) false) m, there exists a monochromatic component on at least false( m + n false) / 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 ≥ 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.

Partitioning a graph into monochromatic connected subgraphs

“…We note that there are examples of 3-colored complete graphs on n vertices with no monochromatic component of order larger than ∕ n 2; hence the lower bound on the order of the monochromatic component is best possible. We present the proof of Proposition 4.3 here for the sake of completeness (see also Corollary 1.9 in DeBiasio et al [4] , contradicting the above. We have thus reached a contradiction to the assumption that | | ∕ B n < 2, which completes the proof.…”

“…, and, thus, the components R and C x ( ) b cover the whole graph. □ Before we turn to prove Conjecture 1.5 for r = 3, we mention the following proposition, which is due to DeBiasio et al [4]. [4]).…”

“…□ Before we turn to prove Conjecture 1.5 for r = 3, we mention the following proposition, which is due to DeBiasio et al [4]. [4]). Let G be a 3-colored graph on n vertices with minimum degree at least ∕ n 7 8.…”

Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

“…Recently, there has been some progress towards Conjecture 3. The best current general result was proved by DeBiasio, Krueger and Sárközy [3].…”

“…Theorem 2 (Gyárfás and Sárközy [6]). Let G be a graph of order n with δ(G) 3 4 n. If the edges of G are 2-coloured, then there exists a monochromatic component of order at least δ(G) + 1.…”