Local Weak Degeneracy of Planar Graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866915267779493888 |
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| author | Davies, Ewan Smith-Roberge, Evelyne |
| author_facet | Davies, Ewan Smith-Roberge, Evelyne |
| contents | 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.). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local Weak Degeneracy of Planar Graphs Davies, Ewan Smith-Roberge, Evelyne Combinatorics 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.). |
| title | Local Weak Degeneracy of Planar Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.21821 |