List-edge-colouring planar graphs with precoloured edges

TL;DR

This paper proves conditions under which a precoloured edge subgraph of a planar graph can be extended to a full edge colouring from lists, depending on maximum degree constraints and list sizes.

Contribution

It establishes new extendability results for list edge-colouring of planar graphs with precoloured edges, depending on subgraph degree and list size.

Findings

Extension is possible if subgraph degree d ≤ t-4 or Δ is large enough.

Extension is impossible for d > t regardless of Δ.

Results depend on maximum degree and list size conditions.

Abstract

Let $G$ be a simple planar graph of maximum degree $\Delta$, let $t$ be a positive integer, and let $L$ be an edge list assignment on $G$ with $|L(e)| \geq \Delta+t$ for all $e \in E(G)$. We prove that if $H$ is a subgraph of $G$ that has been $L$-edge-coloured, then the edge-precolouring can be extended to an $L$-edge-colouring of $G$, provided that $H$ has maximum degree $d\leq t$ and either $d \leq t-4$ or $\Delta$ is large enough ($\Delta \geq 16+d$ suffices). If $d>t$, there are examples for any choice of $\Delta$ where the extension is impossible.