$k$-$L(2,1)$-Labelling for Planar Graphs is NP-Complete for $k\geq 4$

Nicole Eggemann 1 Frédéric Havet 2 Steven Noble 1
2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : A mapping from the vertex set of a graph $G=(V,E)$ into an interval of integers $\{0, \dots ,k\}$ 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 $k\ge 4$, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for $k\leq 3$. For even $k\geq 8$, it remains NP-complete when restricted to planar graphs. % planar graphs for $k\geq 8$. In this paper, we show that it remains NP-complete for any $k \ge 4$ 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.
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[Research Report] RR-6840, INRIA. 2009
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Nicole Eggemann, Frédéric Havet, Steven Noble. $k$-$L(2,1)$-Labelling for Planar Graphs is NP-Complete for $k\geq 4$. [Research Report] RR-6840, INRIA. 2009. 〈inria-00360505〉

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