Asymptotically good local list edge colourings

Abstract

We study list edge colourings under local conditions. Our main result is an analogue of Kahn's theorem in this setting. More precisely, we show that, for a simple graph $G$ with sufficiently large maximum degree $\Delta$ and minimum degree $\delta \geq \ln^{25} \Delta$, the following holds. Suppose that lists of colours $L(e)$ are assigned to the edges of $G$, such that, for each edge $e=uv$, $$|L(e)| \geq (1+o(1)) \cdot \max\left\{\deg(u),\deg(v)\right\}.$$ Then there is an $L$-edge-colouring of $G$. We also provide extensions of this result for hypergraphs and correspondence colourings, a generalization of list colouring.