Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

An Efficient Interference Aware Partially Overlapping Channel Assignment and Routing in Wireless Mesh Networks

In recent years, multi-channel multi-radio wireless mesh networks are considered a reliable and cost effective way for internet access in wide area. A major research challenge in this network is, selecting a least interference channel from the available channels, efficiently assigning a radio to the selected channel, and routing packets through the least interference path. Many algorithms and methods have been developed for channel assignment to maximize the network throughput using orthogonal channels. Recent research and test-bed experiments have proved that POC (Partially Overlapped Channels) based channel assignment allows significantly more flexibility in wireless spectrum sharing. In this paper, first we represent the channel assignment as a graph edge coloring problem using POC. The signal-to-noise plus interference ratio is measured to avoid interference from neighbouring transmissions, when a channel is assigned to the link. Second we propose a new routing metric called signal-to-noise plus interference ratio (SINR) value which measures interference in each link and routing algorithm works based on the interference information. The simulation results show that the channel assignment and interference aware routing algorithm, proposed in this paper, improves the network throughput and performance

The chromatic index of simple graphs

The object of this thesis is twofold: (i) to study the structural properties of graphs which are critical with respect to edge-colourings; (ii) to apply the results obtained to the classification problem arising from Vizing's Theorem. Chapter 1 contains a historical, non-technical introduction, general graph-theoretic definitions and notation, a discussion of Vizing's Theorem as well as a survey of the main results obtained to date in Vizing's classification problem. Chapter 2 introduces the notion of criticality in the first section; the second section contains both well-known and new constructions of critical graphs which will be used in later chapters. The third and final section contains new results concerning elementary properties of critical graphs. Chapter 3 deals with uniquely-colourable graphs and their relationship to critical graphs. Chapter 4 contains results on the connectivity of critical graphs, whereas Chapter 5 deals with bounds on the number of edges of these graphs. In particular, bounds improving those given by Vizing are presented. These results are applied to problems concerning planar graphs. In Chapter 6, critical graphs of small order are discussed. All such graphs of order at most 8 are determined, while the 'critical graph conjecture’ of Beineke & Wilson and Jakobsen is shown to be true for all graphs on at most 10 vertices. The seventh and final chapter deals with circuit length properties of critical graphs. In particular, the minimal order of certain critical graphs with given girth and maximum valency is determined. Results improving Vizing’s estimate of the circumference of critical graphs are also given. The Appendix includes a computer programme which generates critical graphs from simpler ones using a constructive algorithm given in Chapter 2

Time complexity analysis of RLS and (1 + 1) EA for the edge coloring problem

The edge coloring problem asks for an assignment of colors to edges of a graph such that no two incident edges share the same color and the number of colors is minimized. It is known that all graphs with maximum degree Δ can be colored with Δ or Δ + 1 colors, but it is NP-hard to determine whether Δ colors are sufficient. We present the first runtime analysis of evolutionary algorithms (EAs) for the edge coloring problem. Simple EAs such as RLS and (1+1) EA efficiently find (2Δ - 1)-colorings on arbitrary graphs and optimal colorings for even and odd cycles, paths, star graphs and arbitrary trees. A partial analysis for toroids also suggests efficient runtimes in bipartite graphs with many cycles. Experiments support these findings and investigate additional graph classes such as hypercubes, complete graphs and complete bipartite graphs. Theoretical and experimental results suggest that simple EAs find optimal colorings for all these graph classes in expected time O(Δℓ2m log m), where m is the number of edges and ℓ is the length of the longest simple path in the graph

Adaptive multi-code assignment for a DS-CDMA ad hoc network

Master'sMASTER OF ENGINEERIN

Coloration de graphes et attribution d'activités dans des quarts de travail

Revue de littérature -- Organisation de la thèse -- Lower bounds and a tabu search algorithm for the minimum deficiency problem -- On a reduction of the interval coloring problem to a series of bandwidth coloring problems -- About equivalent interval colorings of weighted graphs -- Une approche de programmation en nombres entiers pour la résolution d'un problème d'horaire -- Discussion générale et conclusion