Edge-group choosability of outerplanar and near-outerplanar graphs

Document Type: Research Paper

Author

Department of Mathematics, Kharazmi University, 15719-14911, Tehran, Iran

10.22108/toc.2020.116355.1633

Abstract

Let $\chi_{gl}(G)$ be the {\it{group choice number}} of $G$. A graph $G$ is called {\it{edge-$k$-group choosable}} if its line graph is $k$-group choosable. The {\it{group-choice index}} of $G$, $\chi'_{gl}(G)$, is the smallest $k$ such that $G$ is edge-$k$-group choosable, that is, $\chi'_{gl}(G)$ is the group chice number of the line graph of $G$, $\chi_{gl}(\ell(G))$. It is proved that, if $G$ is an outerplanar graph with maximum degree $\D<5$, or if $G$ is a $({K_2}^c+(K_1 \cup K_2))$-minor-free graph, then $\chi'_{gl}(G)\leq \D(G)+1$. As a straightforward consequence, every $K_{2,3}$-minor-free graph $G$ or every $K_4$-minor-free graph $G$ is edge-$(\D(G)+1)$-group choosable. Moreover, it is proved that if $G$ is an outerplanar graph with maximum degree $\D\geq 5$, then $\chi'_{gl}(G)\leq \D$.

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