Five-list-coloring graphs on surfaces I. Two lists of size two in planar graphs

Let G be a plane graph with outer cycle C, let v(1), v(2) is an element of V(C) and let (L(v) : v is an element of V (G)) be a family of sets such that vertical bar L(v(1))vertical bar = vertical bar L (v(2))vertical bar = 2, vertical bar L(v)vertical bar >= 3 for every v is an element of V(C)\{v(1), v(2)} and vertical bar L(v)vertical bar >= 5 for every v is an element of V (G) \ V (C). We prove a conjecture of Hutchinson that G has a (proper) coloring phi such that phi(v) is an element of L(v) for every v is an element of V (G). We will use this as a lemma in subsequent papers. (C) 2014 The Authors. Published by Elsevier Inc. All rights reserved.