Short Proofs of Rainbow Matchings 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 asymptotically solve some well-known conjectures and to obtain short proofs of old results. In particular: $\bullet $ We give the first asymptotic proof of the “non-bipartite” Aharoni–Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky, and Zerbib. $\bullet $ 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. $\bullet $ We give the first asymptotic proof of a 30-year-old conjecture of Alspach. $\bullet $ We give a simple proof of Pokrovskiy’s asymptotic version of the Aharoni–Berger conjecture with greatly improved error term.

Rainbow independent sets in certain classes of graphs

“…There are many asymptotic results around Conjectures 2.10 and 2.12. Two recent interconnected results were proved in [14]: Some more asymptotic results are quoted in [14]. Caccetta and Mardiyono [11], and independently Chung (see [20]) offered the following: Conjecture 2.17. n Hamiltonian cycles on a vertex set of size n have a rainbow perfect matching.…”

“…There are many asymptotic results around Conjectures 2.10 and 2.12. Two recent interconnected results were proved in [14]:…”