Edge-group choosability of outerplanar and near-outerplanar graphs

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 $D<۵$, or if $G$ is a $({K_۲}^c+(K_۱ \cup K_۲))$-minor-free graph, then $\chi'_{gl}(G)\leq D(G)+۱$. As a straightforward consequence, every $K_{۲,۳}$-minor-free graph $G$ or every $K_۴$-minor-free graph $G$ is edge-$(D(G)+۱)$-group choosable. Moreover, it is proved that if $G$ is an outerplanar graph with maximum degree $D\geq ۵$, then $\chi'_{gl}(G)\leq D$.