Short proofs of rainbow matching results

Abstract

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that "every edge coloured graph of a certain type contains a rainbow matching using every colour". In this paper we introduce a versatile "sampling trick", which allows us to obtain short proofs of old results as well as to solve asymptotically some well known conjectures.• We give a simple proof of Pokrovskiy's asymptotic version of the Aharoni-Berger conjecture with greatly improved error term.• We give the first asymptotic proof of the "non-bipartite" Aharoni-Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky and Zerbib.• We give a very short asymptotic proof of Grinblat's conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat's problem as a function of edge multiplicity of the corresponding multigraph.• We give the first asymptotic proof of a 30 year old conjecture of Alspach.

Proof of Grinblat's conjecture on rainbow matchings in multigraphs

“…Recently, the two authors together with Pokrovskiy [16] showed that v m ≤ 2n + 2m + O(n/(log n) 1/4 ). This substantially improves the result in [17] and is asymptotically tight for m = εn and log −1/4 n ≪ ε ≪ 1.…”

“…Our first step in the proof of Theorem 1.2 will be in this direction. We will show that in any (n, v)multigraph with multiplicity at most (1− δ)n and with v close enough to 3n, we can always find a rainbow matching of size n. In order to prove this, we will first show that one can find a rainbow matching of size n − f (δ), which is a quick corollary of Lemmas 2.9 and 2.10, and then use the sampling trick introduced in [16] to transform this into a result giving a full rainbow matching.…”

“…The first crucial thing to observe is that the Grinblat problem behaves differently when one restricts the edge-multiplicity of the (n, v)-multigraph. For example, it was shown in [16] (improving upon results from [17]) that any (n, 2n + 2m + o(n))-multigraph with edge-multiplicity at most m contains a rainbow matching using all the colours. This indicates that in general, lower multiplicity will make it easier to find such a rainbow matching.…”

“…We can now use the sampling trick introduced in [16] to transform the proposition above into the following theorem.…”

“…Note that indeed this implies the Ryser-Brualdi-Stein conjecture since any properly n-edge-coloured K n,n can be transformed into such a graph by adding to it a disjoint edge repeated in each one of the n colours. This problem has been extensively studied (see, e.g., [2,3,5,13,20]) and the conjecture was shown to hold asymptotically in [19] (see also [16] for a very short proof).…”