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SUMMARY:Seog-Jin Kim (김석진)\, The square of every subcubic planar graph of girth at least 6 is 7-choosable
DESCRIPTION:The square of a graph $G$\, denoted $G^2$\, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Wegner’s conjecture (1977) states that for a planar graph $G$\, the chromatic number $\chi(G^2)$ of $G^2$ is at most 7 if $\Delta(G) = 3$\, at most $\Delta(G)+5$ if $4 \leq \Delta(G) \leq 7$\, and at most $\lfloor \frac{3 \Delta(G)}{2} \rfloor$ if $\Delta(G) \geq 8$. Wegner’s conjecture is still wide open. The only case for which we know tight bound is when $\Delta(G) = 3$. Thomassen (2018) showed that $\chi(G^2) \leq 7$ if $G$ is a planar graph with $\Delta(G) = 3$\, which implies that Wegner’s conjecture is true for $\Delta(G) = 3$. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph\, where $\chi_{\ell}(G^2)$ is the list chromatic number of $G^2$. Cranston and Kim (2008) showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. This is joint work with Xiaopan Lian (Nankai University).
URL:https://dimag.ibs.re.kr/event/2023-09-12/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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