Some applications of graph theory

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.For the abstract of this thesis, please see the attached PDF.This work was funded by a Marie Curie Early Stage Training Fellowship (NET-ACE-programme) under grant number MEST-CT-2004-6724

Non-searchability of random scale-free graphs

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

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 = 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.Comment: 16 pages, includes figures generated using PSTricks. To appear in Discrete Applied Mathematics. Some very minor corrections incorporate

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

International audienceA 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. 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

Non-searchability of random power-law graphs

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