List colourings of graphs

A list colouring of a graph is a colouring in which each vertex υ receives a colour from a prescribed list L(υ) of colours. This paper about list colourings can be thought of as being divided into two parts. The first part, comprising Sections 1, 2 and 6, is about proper colourings, in which adjacent vertices must receive different colours. It is a survey of known conjectures and results with few proofs, although Section 6 discusses several different methods of proof. Section 1 is intended as a first introduction to the concept of list colouring, and Section 2 discusses conjectures and results, mainly about graphs for which “ch = X”. The other part of the paper, comprising Sections 3, 4 and 5, is about improper or defective colourings, in which a vertex is allowed to have some neighbours with the same colour as itself, but not too many. Although still written mainly as a survey, this part of the paper contains a number of new proofs and new conjectures. Section 3 is about subcontractions, and includes conjectures broadly similar to Hadwiger's conjecture. Section 4 is about planar and related graphs. Section 5 is also about planar and related graphs, but this time with additional constraints imposed on the lists.