Edge-colouring graphs with local list sizes

TL;DR

This paper extends Kahn's asymptotic result on list edge-colouring to a local setting, showing that graphs with large degrees and certain degree conditions can be edge-coloured from local lists.

Contribution

It provides a local generalization of Kahn's theorem, applying to graphs with large degrees and local list size conditions, and extends to hypergraphs and correspondence colouring.

Findings

Proves local version of Kahn's asymptotic list colouring theorem.

Shows graphs with large maximum degree and high minimum degree are list colourable from local lists.

Extends results to hypergraphs and correspondence colouring through a weighted approach.

Abstract

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph $G$ with sufficiently large maximum degree $\Delta$ and minimum degree $\delta \geq \ln^{25} \Delta$, the following holds: for every assignment of lists of colours to the edges of $G$, such that $|L(e)| \geq (1+o(1)) \cdot \max\left\{\rm{deg}(u),\rm{deg}(v)\right\}$ for each edge $e=uv$, there is an $L$-edge-colouring of $G$. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, $k$-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.