k-L(2,1)-labelling for planar graphs is NP-complete for k≥4

Authors:
Nicole Eggemann;Frédéric Havet;Steven D. Noble
Affiliations:
Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK;Projet Mascotte, I3S (CNRS and University of Nice-Sophia-Antipolis), France and INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France;Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK
Venue:
Discrete Applied Mathematics
Year:
2010

Citing 14
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Abstract

A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices at distance 2 are mapped onto distinct integers. It is known that, for any fixed k=4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k@?3. For even k=8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k=4 by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this.