Title:

Listcolourings of nearouterplanar graphs

A listcolouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of listcolouring, each for the class of a nearouterplanar graphs. Since a graph is outerplanar if it is both K4minorfree and K2,3minorfree, then by a nearouterplanar graph we mean a graph that is either K4minorfree or K2,3minorfree. Chapter 1 gives an introduction to the area of graph colourings, and includes a review of several results and conjectures in this area. In particular, four important and interesting conjectures in graph theory are the ListEdgeColouring Conjecture (LECC), the ListTotalColouring Conjecture (LTCC), the Entire Colouring Conjecture (ECC), and the ListSquareColouring Conjecture (LSCC), each of which will be discussed in Chapter 1. In Chapter 2 we include a proof of the LECC and LTCC for all nearouterplanar graphs. In Chapter 3 we will study the listcolouring of a nearouterplanar graph in which vertices and faces, edges and faces, or vertices, edges and face are to be coloured. The results for the case when all elements are to be coloured will prove the ECC for all nearouterplanar graphs. In Chapter 4 we will study the listcolouring of the square of a K4minorfree graph, and in Chapter 5 we will study the listcolouring of the square of a K2,3minorfree graph. In Chapter 5 we include a proof of the LSCC for all K2,3minorfree graphs with maximum degree at least six.
