Bipartite rainbow numbers of matchings

Abstract

Given two graphs G and H , let f ( G , H ) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least r b ( G , H ) = f ( G , H ) + 1 colors contains a rainbow copy of H , where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number r b ( G , H ) is called the rainbow number of H with respect to G , and simply called the bipartite rainbow number of H if G is the complete bipartite graph K m , n . Erdős, Simonovits and Sós showed that r b ( K n , K 3 ) = n . In 2004, Schiermeyer determined the rainbow numbers r b ( K n , K k ) for all n ≥ k ≥ 4 , and the rainbow numbers r b ( K n , k K 2 ) for all k ≥ 2 and n ≥ 3 k + 3 . In this paper we will determine the rainbow numbers r b ( K m , n , k K 2 ) for all k ≥ 1 .