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 with a common neighbour 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 = 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.Comment: 16 pages, includes figures generated using PSTricks. To appear in
Discrete Applied Mathematics. Some very minor corrections incorporate
