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

  • Authors:

  • Zs. Tuza;M. Voigt

  • Affiliations:

  • Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary and Department of Computer Science, University of Veszprem, Hungary;Institut für Mathematik, Technische Universität Ilmenau, Postfach 0565, D-98684 Ilmenau, Germany

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

Quantified Score

Hi-index 0.05

Visualization

Abstract

An L-list colouring of a graph G is a proper vertex colouring in which every vertex υ gets a colour from a prescribed list L(υ) 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(υ) to the vertices of G with |L(υ)| = 3 for υ ∈ W and |L(υ)| = 5 otherwise, such that an L-list colouring is not possible. The existence of such graphs is in sharp contrast with Thomassen's 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).