Large Rainbow Matchings in Edge-Coloured Graphs

Abstract

<jats:p>A<jats:italic>rainbow subgraph</jats:italic>of an edge-coloured graph is a subgraph whose edges have distinct colours. The<jats:italic>colour degree</jats:italic>of a vertex<jats:italic>v</jats:italic>is the number of different colours on edges incident with<jats:italic>v</jats:italic>. Wang and Li conjectured that for<jats:italic>k</jats:italic>≥ 4, every edge-coloured graph with minimum colour degree<jats:italic>k</jats:italic>contains a rainbow matching of size at least ⌈<jats:italic>k</jats:italic>/2⌉. A properly edge-coloured<jats:italic>K</jats:italic><jats:sub>4</jats:sub>has no such matching, which motivates the restriction<jats:italic>k</jats:italic>≥ 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊<jats:italic>k</jats:italic>/2⌋ is guaranteed to exist, and they proved several sufficient conditions for a matching of size ⌈<jats:italic>k</jats:italic>/2⌉. We prove the conjecture in full.</jats:p>

Rainbow edge-coloring and rainbow domination

“…LeSaulnier, Stocker, Wenger, and West [11] showed that alwaysα (G) ≥ δ (G)/2 . Exploiting this result and its lemmas, Kostochka and Yancey [9] proved the conjecture of Wang and Li completely:…”

“…We consider t-tolerant edge-colored graphs with n vertices. In Section 2 we construct examples withχ (G) ≥ t 2 (n−1) and prove that alwaysχ (G) < t(t + 1)n ln n. There is potential for improving the upper bound: Kostochka, Pfender, and Yancey [10] showed further that if |V (G)| > 5.5δ(G) 2 , then G has a rainbow matching of size at leastδ(G).…”

Rainbow matchings in strongly edge-colored graphs

“…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.…”

“…From a result of Kostochka and Yancey [5] for arbitrary edge-colored graphs, it follows that if G is a properly edge-colored graph that is not K 4 , then G contains a rainbow matching of size…”

“…Wang [10] showed that if G is a properly edge-colored graph with |V (G)| ≥ 8δ(G) 5 , then G has a rainbow matching of size at least  3δ(G) 5…”