An upper bound for the total chromatic number

The total chromatic number, Z"(G), of a graph G, is defined to be the minimum number ofcolours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that )('(G) _< z'(G) + 2x/~G), where z(G)is the vertex chromatic number and )((G)is the edge chromatic number of the graph. A (proper) total colouring of a graph is the assignment of colours to the vertices and edges of a graph in such a way that no two adjacent edges, no two adjacent vertices and no incident vertex and edge are given the same colour. The total chromatic number, Z"(G), of a graph G is defined to be the minimum number of colours that are needed to produce a total colouring of G. The concept of a total colouring was introduced by Behzad in 1965 [1], he conjectured that any graph of maximum degree A(G) has a total chromatic number satisfying z'(G) < A(G) + 2. This conjecture is known as the total colouring conjecture (TCC). Several types of graph have been shown to satisfy the TCC; including complete graphs and bipartite graphs [2], complete k-partite graphs [3], all graphs of maximum degree at most three [8], and all graphs of maximum degree equal to four [6]. The proof required for this last class is unexpectedly involved. In a proper total colouring different colours must be assigned to a vertex and its incident edges, thus Z"(G) -> A(G) + 1. Bollob/ts and Harris [5] have shown, as a corollary to a result for list colourings, that )('(G) _< ~A(G) for large A(G) and Kostocka [7] has shown that )('(G) _< ~A(G)for A(G) _> 6. No better bounds appear to be known. As the main result of this paper we show that )('(G) = A(G) + o(A). The notation used in this paper is standard (see [4]). All graphs considered will be finite graphs. We need only consider those connected graphs which are neither an odd cycle nor a complete graph. A colouring of a set C with k colours is a map ¢: C , {1,2,.. . ,k}. A colouring can also be considered as a partition (C1, C 2 . . . . , Ck) where C i is the set of elements coloured with colour i. We consider colourings of the sets V(G), E(G), * Partially supported by ORS grant ORS/84120