Simultaneous edge-colourings

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 $\hat{I}”$, 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 $\sqrt{k}\hat{I}”+o(\hat{I}”)$ colours always suffice, which tends to the optimal value as $k$ grows. We also show that $\ell \hat{I}”+o(\hat{I}”)$ 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 Füredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings.

14 pages