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 Fig. 1). Suppose that each edge of this graph is assigned a list of 3 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 are the lists assigned to the edges of K4. The problem is a special case of a long-standing conjecture in graph theory, known as the List-Colouring Conjecture (hereafter abbreviated as “LCC”), first posed by Vizing in 1976 [4]. Before stating the LCC and proving the above mentioned result we shall need to give a few definitions. A graph G consists, for our purposes, of a finite set V = {v1, v2, . . . , vn} of vertices together with a set E of unordered pairs of distinct elements of V , called edges. If e is an edge corresponding to the unordered pair {x, y}, we also denote e by xy or yx. The edge xy is said to be incident to or to join the vertices x and y. The order of a graph is its number of vertices. A graph G is usually represented pictorially by drawing vertices as points (or, more precisely, little circles) and edges by (straight or curved) lines joining the corresponding pairs of vertices, so that no line representing an edge intersects any point representing a vertex other than the two points which the line itself is incident to (see Fig. 1). A graph G is complete if every pair of vertices of G is joined by an edge. A complete graph of order n will be denoted by Kn. A cycle (of length n) is a graph G whose vertex set is {v1, v2, v3, . . . , vn−1, vn} and whose edge set is {v1v2, v2v3, . . . , vn−1vn, vnv1}. We now introduce some definitions from graph colouring. Let C be a set (to be thought of as the set of “colours”), and let G be a graph with edge set E. An edge-colouring of G with colour set C is a function φ : E → C which satisfies the property that φ(e1) 6= φ(e2) for every pair of distinct edges which are incident with the same vertex. Clearly the