Rainbow Matchings of Size $\delta(G)$ in Properly Edge-Colored Graphs

Abstract

A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f (δ) such that a properly edgecolored graph G with minimum degree δ and order at least f (δ) must have a rainbow matching of size δ. We answer this question in the affirmative; f (δ) = 6.5δ suffices. Furthermore, the proof provides a O(δ(G)|V (G)| 2 )-time algorithm that generates such a matching.

Rainbow matchings and cycle-free partial transversals of Latin squares

“…Positive answers to Problem 1 were given in [10], [5], [6], the current best bound is 98δ(G) 23 in [6]. In this paper we give a better bound, namely 4δ(G) − 3.…”

Large Rainbow Matchings in Edge-Coloured Graphs

“…By the bipartite nature of G 0 , we may assume that y ∈ L. By Claim 4, w 1 (y) − w 1 (xy) ≤ k−1 2 . Because Options (1) and (2) were not performed at Step i, w 1 (x) + w 1 (E G i−1 [φ(xy)]) ≤ 4(k − i). Therefore the weight of Step i is at most k−1 2 + 4(k − i) < 4.5k − 4i.…”

“…Diemunsch et al [2] answered the question in the positive and proved that f (k) ≤ 6.5k. Shortly thereafter, Lo [12] improved the bound to f (k) ≤ 4.5k, and finally Diemunsch et al [3] combined the two manuscripts and improved the bound to f (k) ≤ 98 23 k. The largest matching in a graph with n vertices contains at most n/2 edges, so f (k) ≥ 2k.…”

“…Several ideas in the proof came from Diemunsch et al's paper [2]. The full proof is presented in the next section.…”

Existences of rainbow matchings and rainbow matching covers

“…Wang [11] asked does there exist a function f (k) such that every properly edge-coloured graph G on n ≥ f (k) vertices with δ(G) ≥ k contains a rainbow matching of size at least k. Diemunsch et al [1] showed that such function does exist and f (k) ≤ 98k/23. Gyárfás and Sarkozy [2] improved the result to f (k) ≤ 4k − 3.…”