Rainbow matchings in edge-colored graphs

TL;DR

This paper investigates the existence of rainbow matchings of size k in edge-colored graphs, establishing tight conditions based on the sum of edges and colors, extending previous results on rainbow triangles and cliques.

Contribution

It introduces a new tight condition for the existence of rainbow matchings of size k in edge-colored graphs, generalizing prior work on rainbow triangles and cliques.

Findings

Established a tight bound on e(G)+c(G) for rainbow matchings

Extended previous results from triangles and cliques to matchings

Provided a generalized condition applicable to all edge-colored graphs

Abstract

Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges and colors in $G$, respectively. A subgraph $H$ is called rainbow if $c(H)=e(H)$. Li et al. (European J. Combin., 36 (2014), 453-459) proved that every edge-colored graph on $n$ vertices with $e(G)+c(G) \geq n(n+1)/2$ contains rainbow triangles. Later, Xu et al. (European J. Combin., 54 (2016), 193-200) generalized the previous results concerning rainbow triangles to rainbow cliques $Kr$, where $r\geq 4$. In this paper, we consider the existence of rainbow matchings of size $k$ in general edge-colored graphs $G$ under the condition of $e(G)+c(G)$, and the condition in our result is tight.