92.80 The edge-choosability of the tetrahedron

92.80 The edge-choosability of the tetrahedron We consider a list-colouring type of problem on the edges of the graph K4, which is the skeleton of a tetrahedron and consists of four vertices, each pair of which is joined by one edge (see Figure 1). Suppose that each edge of this graph is assigned a list of three distinct colours, where the list assigned to each edge may be different from the lists assigned to the other edges. The question which we address is the following: is it possible to choose one colour for each edge from the edge's list in such a way that the resulting colouring of the edges of K4 has the property that any two edges which are adjacent (i.e. have a common endpoint) receive a different colour? We shall see that the answer to this question is always positive, no matter which lists are assigned to the edges of K4. The problem is a special case of a longstanding conjecture in graph theory, known as the List-Colouring Conjecture (LCC), first posed by Vizing in 1976 [1]. Before stating the LCC and proving the above-mentioned result we shall need to give a few definitions.