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Mathematics > Combinatorics

arXiv:0909.2613 (math)
[Submitted on 14 Sep 2009 (v1), last revised 30 Jul 2010 (this version, v2)]

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

Authors:Nicole Eggemann, Frédéric Havet, Steven D. Noble
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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 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.
Comments: 16 pages, includes figures generated using PSTricks. To appear in Discrete Applied Mathematics. Some very minor corrections incorporated
Subjects: Combinatorics (math.CO)
MSC classes: 05C15 (Primary) 68R10 (Secondary)
Report number: inria-00360505
Cite as: arXiv:0909.2613 [math.CO]
  (or arXiv:0909.2613v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.0909.2613
Related DOI: https://doi.org/10.1016/j.dam.2010.06.016

Submission history

From: Steven Noble [view email]
[v1] Mon, 14 Sep 2009 18:36:35 UTC (21 KB)
[v2] Fri, 30 Jul 2010 14:31:34 UTC (20 KB)
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