Local girth choosability of planar graphs

TL;DR

This paper introduces a new local girth list assignment concept for planar graphs, unifying and extending Thomassen's results on choosability related to girth, with a theorem ensuring list-colorability under these local conditions.

Contribution

It presents a unifying local list coloring theorem for planar graphs based on shortest cycle length, generalizing Thomassen's classical results.

Findings

If each vertex has at least 3 colors, the graph is list-colorable.

Vertices in 4-cycles require at least 4 colors.

Vertices in triangles require at least 5 colors.

Abstract

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a \emph{local girth list assignment}: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if $G$ is a planar graph and $L$ is a list assignment for $G$ such that $|L(v)| \geq 3$ for all $v \in V(G)$; $|L(v)| \geq 4$ for every vertex $v$ contained in a 4-cycle; and $|L(v)| \geq 5$ for every $v$ contained in a triangle, then $G$ admits an $L$-colouring.