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

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. (C) 2010 Elsevier B.V. All rights reserved.