Rainbow matchings in edge-colored simple graphs

Abstract

There has been much research on the topic of finding a large rainbow matching (with no two edges having the same color) in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barát, Gyárfás, and Sárközy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not loops) with 2q colors where each color appears at least q times, there is always a rainbow matching of size q. Recently, Aharoni, Berger, Chudnovsky, Howard, and Seymour proved a relaxation of the conjecture with 3q − 2 colors. Our main result proves that 2q + o(q) colors are enough if the graph is simple, confirming the conjecture asymptotically for simple graphs. This question restricted to simple graphs was considered before by Aharoni and Berger. We also disprove one of their conjectures regarding the lower bound on the number of colors one needs in the conjecture of Barát, Gyárfás, and Sárközy for the class of simple graphs. Our methods are inspired by the randomized algorithm proposed by Gao, Ramadurai, Wanless, and Wormald to find a rainbow matching of size q in a graph that is properly edge-colored with q colors, where each color class contains q + o(q) edges. We consider a modified version of their algorithm, with which we are able to prove a generalization of their statement with a slightly better error term in o(q). As a by-product of our techniques, we obtain a new asymptotic version of the Brualdi-Ryser-Stein Conjecture.

Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings

“…Already though, Theorem 2.11 generalizes the main results of Gao, Ramadurai, Wanless, and Wormald [22] (their Theorems 1.4 and 1.7) to allowing any number of colors (the sparse setting) while also allowing larger multiplicity; it thus almost proves a conjecture of Aharoni and Berger [1] (their Conjecture 2.5) for bipartite 3-uniform hypergraphs but with an additional multiplicity assumption (the conjecture is actually not true without some additional assumption) and also generalizes it to all uniformities. Similarly, Theorem 2.11 generalizes two of the main results of Charkaborti and Loh [11] (their Theorems 1.10 and 1.12) to the sparse setting, hypergraphs and with better multiplicity assumptions.…”

“…Conjecture 2.17 (Chakraborti and Loh [11]). For every real ε > 0, there exists q 0 such that for all integers q ≥ q 0 the following holds: If G is a bipartite simple graph that is properly edge colored with (1 + ε)q colors such that each color appears exactly q times, then G has a rainbow matching of size q.…”