Rainbow matchings in edge-colored graphs

TL;DR

This work determines a tight bound on the sum of the number of edges and colors, $e(G) + c(G)$, in an edge-colored graph that guarantees a rainbow matching of size $k$. By constructing a rainbow subgraph $G^r$ with one edge per color and applying an Erdős–Gallai–type decomposition, the authors perform a meticulous case analysis (perfect and non-perfect matching scenarios) to show that any counterexample would violate the bound. They also provide extremal constructions illustrating the bound's tightness. The results extend rainbow-extremal phenomena from triangles and cliques to matchings, contributing to the anti-Ramsey theory of rainbow structures in 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.

Theorems & Definitions (34)

• Theorem 1: Erdős and Gallai, Erd2
• Theorem 2
• Theorem 3: Lovász and Plummer, Lov
• Theorem 4: Li, Tu, and Jin, Li
• Theorem 5: Dirac, Dir
• Claim 1
• proof
• Claim 2
• proof
• Lemma 6: Chen, Li, and Tu, Lemma 3.5 in Che