Simultaneous edge-colourings

TL;DR

This paper investigates the problem of simultaneously edge-colouring multiple graphs with minimal colours, providing asymptotically optimal bounds and confirming several conjectures in graph theory.

Contribution

It introduces new bounds for simultaneous edge-colouring, confirming conjectures for small and infinite values of k, and extends results to list colourings and related problems.

Findings

Asymptotically optimal bounds for small k

Confirmation of Cabello's conjecture for k=2

Extension to list colourings and related conjectures

Abstract

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the maximum degree $\Delta$, for small values of $k$ and for an infinite sequence of values of $k$. This asymptotically settles a conjecture of Cabello for $k=2$. Moreover, we show that $\sqrt k \Delta + o(\Delta)$ colours always suffice, which tends to the optimal value as $k$ grows. We also show that $\ell \Delta + o(\Delta)$ colours are enough when every edge appears in at most $\ell$ of the graphs, which asymptotically confirms a conjecture of Cambie. Finally, our results extend to the list setting. We also find a close connection to a conjecture of F\"uredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings.