A Note on Large Rainbow Matchings in Edge-coloured Graphs

Abstract

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on n vertices with minimum colour degree at least k contains a rainbow matching of size at least k, provided n ≥ 17 4 k 2 . In this paper, we show that n ≥ 4k − 4 is sufficient for k ≥ 4.

Orthogonal matchings revisited

“…Recently, the study of rainbow paths and cycles under minimum color degree condition has received much attention, see [6,15]. For rainbow matchings under minimum color degree condition, see [11,10,16,13,14,19].…”

Rainbow matchings in strongly edge-colored graphs

“…Wang (2011) asked whether there exists a function f such that a properly edge-colored graph G with at least f (δ(G)) vertices is guaranteed to contain a rainbow matching of size δ(G). This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is f (k) = 4k − 4, for k ≥ 4 and f (k) = 4k − 3, for k ≤ 3. Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than 4δ(G) − 4 vertices.…”

“…We show that for a strongly edge-colored graph G, if |V (G)| solvable, deciding whether an edge-colored graph has a maximum rainbow matching of size at least k is an NP-Complete problem, mentioned in Garey and Johnson [2] as the Multiple Choice Matching problem. There have been several studies giving lower bounds for the size of maximum rainbow matchings in edge-colored graphs [11,6,5,7]. Rainbow matchings in properly edge-colored graphs were studied in connection with the famous Latin square transversal problem.…”

“…Several positive answers were obtained for this question, and Gyárfás et al [3] showed that every properly edge-colored graph G with |V (G)| ≥ 4δ(G) − 3 contains a rainbow matching of size δ(G). A result by Lo and Tan [7] for arbitrary edge-colored graphs implies that if δ(G) ≥ 4, then any properly edgecolored graph G with |V (G)| ≥ 4δ(G) − 4 contains a rainbow matching of size δ(G).…”

“…It would be also interesting to show that some number of vertices less than 4δ(G)−4 suffices to guarantee a rainbow matching of size δ(G) in a strongly edge-colored graph with δ(G) ≥ 4, improving on the threshold from [7] for properly edge-colored graphs.…”