A note on planar 5-list colouring: non-extendability at distance 4 Original Research Article

An L-list colouring of a graph G is a proper vertex colouring in which every vertex v gets a colour from a prescribed list L(v) of allowed colours. Albertson has posed the following problem: Suppose G is a planar graph and each vertex of G has been assigned a list of five colours. Let W⊆V(G) such that the distance between any two vertices of W is at least d (=4). Can any list colouring of W be extended to a list colouring of G ? We give a construction satisfying the assumptions for d=4 where the required extension is not possible. As an even stronger property, in our example one can assign lists L(v) to the vertices of G with |L(v)|=3 for v∈W and |L(v)|=5 otherwise, such that an L-list colouring is not possible. The existence of such graphs is in sharp contrast with Thomassenʹs theorem stating that a list colouring is always possible if the vertices of 3-element lists belong to the same face of G (and the other lists have 5 colours each).