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

Optimal L(h, k)-labeling of regular grids

The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h, k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h, k)-labeling problem has been intensively studied in some special cases, i.e. when k = 0 (vertex coloring), h = k (vertex coloring the square of the graph) and h = 2k (radio- or λ-coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h, k)-labeling problem on cellular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds. For the sake of completeness, we study also hexagonal and squared grids. © 2006 Discrete Mathematics and Theoretical Computer Science (DMTCS)

Conflict-free star-access in parallel memory systems

We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should: (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates. i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadth-first search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the star-template access problem on two specific host graphs-tori and hypercubes-that are also popular interconnection network topologies. The proposed conflict-free mappings on these graphs are fast, use an optimal or provably good number of memory modules, and guarantee load balancing. (C) 2006 Elsevier Inc. All rights reserved

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

Labelling of Cactus Graphs

The -labelling of a graph is an abstraction of assigning integer frequencies to radio transmitters such that the transmitters that are one unit of distance apart receive frequencies that differ by at least two, and transmitters that are two units of distance apart receive frequencies that differ by at least one. The span of an -labelling is the difference between the largest and the smallest assigned frequency. The -labelling number of a graph , denoted by , is the least integer such that has an -labelling of span . A cactus graph is a connected graph in which every block is either an edge or a cycle. The goal of the problem is to show that for a cactus graph , where is the degree of . An optimal algorithm is also presented here to label the vertices of cactus graph using -labelling technique in time, where is the total number of vertices of the cactus graph