Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K-4-minor-free graph with maximum degree Delta, then G is entirely 7-choosable if Delta <= 4 and G is entirely (Delta + 2)-choosable if Delta >= 5: that is, if every vertex, edge and face of G is given a list of max{7, Delta + 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 K-2,K-3-minor-free graph or a ((K) over bar (2) + (K-1 boolean OR K-2))-minor-free graph. As a special case this proves that the Entire Coluring Conjecture, that a plane graph is entirely (6 + 4)colourable, holds if C is a plane embedding of a K4-minor-free graph, a K-2,K-3-minor-free graph or a ((K) over bar (2) + (K-1 boolean OR K-2))-minor-free graph. (C) 2008 Elsevier B.V. All rights reserved.