Proof of Grinblat's conjecture on rainbow matchings in multigraphs

TL;DR

This paper proves Grinblat's conjecture that large enough multigraphs with specific colour and clique conditions always contain a rainbow matching of size n, advancing understanding in combinatorial rainbow structures.

Contribution

The paper confirms Grinblat's conjecture for all sufficiently large n, establishing the existence of large rainbow matchings in complex multigraphs.

Findings

Proves Grinblat's conjecture for large n.

Establishes conditions for rainbow matchings in multigraphs.

Advances combinatorial theory on rainbow structures.

Abstract

Many well-known problems in Combinatorics can be reduced to finding a large rainbow structure in a certain edge-coloured multigraph. Two celebrated examples of this are Ringel's tree packing conjecture and Ryser's conjecture on transversals in Latin squares. In this paper, we answer such a question raised by Grinblat twenty years ago. Let an $(n,v)$-multigraph be an $n$-edge-coloured multigraph in which the edges of each colour span a disjoint union of non-trivial cliques that have in total at least $v$ vertices. Grinblat conjectured that for all $n \geq 4$, every $(n,3n-2)$-multigraph contains a rainbow matching of size $n$. Here, we prove the conjecture for all sufficiently large $n$.