Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2108.07734

Short proofs of rainbow matching results

David Munhá Correia·Alexey Pokrovskiy·Benny Sudakov
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David Munhá Correia, Alexey Pokrovskiy, Benny Sudakov. 2021-09-02. Short proofs of rainbow matching results. https://arxiv.org/abs/2108.07734

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗