edge-group choosability of outerplanar and near-outerplanar graphs

Let χgl(g) be the group choice number of g . a graph g is called edge-k -group choosable if its line graph is k -group choosable. the group-choice index of g , χ′gl(g) , is the smallest k such that g is edge-k -group choosable, that is, χ′gl(g) is the group chice number of the line graph of g , χgl(ℓ(g)) . it is proved that, if g is an outerplanar graph with maximum degree d<5 , or if g is a (k2c+(k1∪k2)) -minor-free graph, then χ′gl(g)≤d(g)+1 . as a straightforward consequence, every k2,3 -minor-free graph g or every k4 -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≥5 , then χ′gl(g)≤d