Rainbow number of matchings in regular bipartite graphs

TL;DR

This paper investigates the minimum number of colors needed in any edge-coloring of regular bipartite graphs to guarantee a rainbow matching of a given size, providing bounds and exact values for various graph classes.

Contribution

It establishes bounds for the rainbow number of matchings in regular bipartite graphs and determines exact values for paths and cycles, advancing understanding of rainbow subgraph existence.

Findings

Bounds for $rb(B_{n,k},mK_2)$ are established.

For large $n$, the rainbow number reaches the lower bound.

Exact rainbow numbers are determined for paths and cycles.

Abstract

Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to $G$. Denote $mK_2$ a matching of size $m$ and $B_{n,k}$ a $k$-regular bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|=n$ and $k\leq n$. In this paper we give an upper and lower bound for $rb(B_{n,k},mK_2)$, and show that for given $k$ and $m$, if $n$ is large enough, $rb(B_{n,k},mK_2)$ can reach the lower bound. We also determine the rainbow number of matchings in paths and cycles.