Local Weak Degeneracy of Planar Graphs

TL;DR

The paper addresses local-girth colouring for planar graphs within a weak degeneracy framework and proves that planar graphs are weakly f-degenerate for the local girth function f with f(v) ≥ max{7 − g(v), 2}, yielding a correspondence-colouring analogue and a unified strengthening of several classical results. The authors develop a Del/DelSave-based inductive method, define canvases to capture boundary and girth constraints, and perform boundary-path reductions while tracking two auxiliary independent-vertex sets A and B. The main contributions extend Thomassen’s 5-list-colourability, the girth≥5 3-list-colourability, and the degeneracy-based weak results to the local-girth and correspondence-colouring setting, with a self-contained, structure-driven proof that avoids certain earlier exponential bounds. The work clarifies the role of outer-face structure and chords in planar graphs and provides a robust approach that could extend to correspondence-painting and related online colouring settings.

Abstract

Thomassen showed that planar graphs are 5-list-colourable, and that planar graphs of girth at least five are 3-list-colourable. An easy degeneracy argument shows that planar graphs of girth at least four are 4-list-colourable. In 2022, Postle and Smith-Roberge proved a common strengthening of these three results: with $g(v)$ denoting the length of a shortest cycle containing a vertex $v$, they showed that if $G$ is a planar graph and $L$ a list assignment for $G$ where $|L(v)| \geq \max\{3,8-g(v)\}$ for all $v \in V(G)$, then $G$ is $L$-colourable. Moreover, they conjectured that an analogous theorem should hold for correspondence colouring. We prove this conjecture; in fact, our main theorem holds in the still more restrictive setting of weak degeneracy, and moreover acts as a joint strengthening of the fact that planar graphs are weakly 4-degenerate (originally due to Bernshteyn, Lee, and Smith-Roberge), and that planar graphs of girth at least five are weakly 2-degenerate (originally due to Han et al.).

Figures (13)

• Figure 1: The exceptional cases of Theorem \ref{['thm:inductive']}. Vertices of the path $P$ are squares.
• Figure 2: Cases for Lemma \ref{['lem:1chord']} in which we argue that the 1-chord $uv$ (the thick edge) cannot be in a minimal counterexample.
• Figure 3: 2-chords $xyz$ of the type that are not ruled out by Lemma \ref{['lem:2chords']}.
• Figure 4: 3-chords $xyzw$ of the type we cannot rule out in Lemma \ref{['lem:3chords']}: $g(y)\ge 5$, neither $x$ nor $w$ is an internal vertex of $P$, and one side of the chord is an exception of type \ref{['ex:A']} or \ref{['ex:AB']}.
• Figure 5: The decomposition of $\delta G$ into $PP'P"u_k$. If $v_0$ exists then $v_0\in A$, and we have $f(v_1)\ge 2$ in any case. The labels $u_k$ and $v_{t+1}$ refer to the same vertex for convenience.

Theorems & Definitions (71)

• Definition 1.1
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