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

TL;DR

This paper proves that the k-L(2,1)-labelling problem remains NP-complete for all k >= 4, even when restricted to planar graphs, by establishing the NP-completeness of a related planar graph problem.

Contribution

It demonstrates NP-completeness of k-L(2,1)-labelling for all k >= 4 on planar graphs and provides a proof of NP-completeness for Planar Cubic Two-Colourable Perfect Matching.

Findings

NP-completeness of k-L(2,1)-labelling for all k >= 4 on planar graphs

Proof of NP-completeness for Planar Cubic Two-Colourable Perfect Matching

Extension of known complexity results to all k >= 4 in planar graphs

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.