k-L(2,1)-Labelling for Planar Graphs is NP-Complete for k >= 4
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 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 <= 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.
Keywords
Cite
@article{arxiv.0909.2613,
title = {k-L(2,1)-Labelling for Planar Graphs is NP-Complete for k >= 4},
author = {Nicole Eggemann and Frédéric Havet and Steven D. Noble},
journal= {arXiv preprint arXiv:0909.2613},
year = {2010}
}
Comments
16 pages, includes figures generated using PSTricks. To appear in Discrete Applied Mathematics. Some very minor corrections incorporated