Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree ∆, then G is entirely 7-choosable if ∆ ≤ 4 and G is entirely (∆+2)-choosable if ∆ ≥ 5; that is, if every vertex, edge and face of G is given a list of max{7,∆+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K2,3-minor-free graph or a (K̄2 + (K1 ∪K2))minor-free graph. As a special case this proves that the Entire Coluring Conjecture, that a plane graph is entirely (∆ + 4)-colourable, holds if G is a plane embedding of a K4-minor-free graph, a K2,3-minor-free graph or a (K̄2+(K1∪K2))-minor-free graph.