Abstract. We model a problem proposed by Alcatel, a satellite building company, using improper colourings of graphs. The relation between improper colourings and maximum average degree is underlined, which contributes to generalise and improve previous known results about improper colourings of planar graphs. 1

This paper aims to provide a detailed survey of existing graph models and algorithms for important problems that arise in di#erent areas of wireless telecommunication. In particular, applications of graph optimization problems such as minimum dominating set, minimum vertex coloring and maximum clique in multihop wireless networks are discussed.

. Let P 10 ne be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P 10 ne. The decomposition is used to show that graphs in this class are 3-edge-colourable. We also consider an application to a conjecture due to Grotzsch which states that a planar graph is 3-edge-colourable if and only if it is fractionally 3-edge-colourable. 1985 Mathematics Subject Classification: 05C50,05C75. Key Words and Phrases: Planar graph, Petersen Graph, Four Colour Problem, cubic, matching polyhedron, integer decomposition property, edge-colouring. 1 Background We consider loopless graphs G = (V; E) with node set V and edge set E. For an edge e 2 E, we denote by G=e the graph obtained by contracting the edge e, i.e., identifying its ends and deleting the resulting loop. A subgraph (respectively induced subgraph) of G is a graph obtained by deleting nodes or edges (respectively deleting nodes) of G. Informally, a...

Contents 1 Introduction 5 1.1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 History of the four-colour problem : : : : : : : : : : : : : : : : : : 6 2 Applications of Euler's formula 11 2.1 Euler's formula : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.2 Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3 Kempe's approach 19 3.1 The first `proof' of the four-colour theorem : : : : : : : : : : : : : 19 3.2 The five-colour theorem : : : : : : : : : : : : : : : : : : : : : : : 21 3.3 A reduction theorem : : : : : : : : : : : : : : : : : : : : : : : : : 22 4 Other approaches to the problem 25 4.1 Hamilton cycles : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4.2 Edge-colourings : : : : : : : : : : : : : : : : : : :

Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 1930, an efficient solution to it was found only in the seventies of the last century. Planar graphs play an important role both in the graph theory and in the graph drawing areas. In fact, planar graphs have several interesting properties: for example they are sparse, four-colorable, allow a number of operations to be performed efficiently, and their structure can be elegantly described by an SPQR-tree (see Section 3.1.2). From the information visualization perspective, instead, as edge crossings turn out to be the main culprit for reducing readability, planar drawings of graphs are considered clear and comprehensible. As a matter of fact, the study of planarity has motivated much of the development of graph theory. In this chapter we review the number of alternative algorithms available in the literature for efficiently testing planarity and computing planar embeddings. Some of these algorithms