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A mapping from the vertex set of a graph G = (V,E) into an interval ofintegers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacentvertices are mapped onto integers that are at least 2 apart, and every twovertices with a common neighbour are mapped onto distinct integers. It is knownthat for any fixed k >= 4, deciding the existence of such a labelling is anNP-complete problem while it is polynomial for k <= 3. For even k >= 8, itremains NP-complete when restricted to planar graphs. In this paper, we showthat it remains NP-complete for any k >= 4 by reduction from Planar CubicTwo-Colourable Perfect Matching. Schaefer stated without proof that PlanarCubic Two-Colourable Perfect Matching is NP-complete. In this paper we give aproof of this.

k-L(2,1)-Labelling for Planar Graphs is NP-Complete for k >= 4