Bipartite rainbow numbers of matchings

Abstract: Given two graphs and , let denote the maximum number for which there is a way to color the edges of with colors such that every subgraph of has at least two edges of the same color. Equivalently, any edge-coloring of with at least colors contains a rainbow copy of , where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number is called the rainbow number of with respect to , and simply called the bipartite rainbow number of if is the complete bipartite graph . Erdős, Simonovits and Sós showed that . In 2004, Schiermeyer determined the rainbow numbers for all , and the rainbow numbers for all and . In this paper we will determine the rainbow numbers for all . [Copyright &y& Elsevier]