The Total Chromatic Number of Complete Multipartite Graphs with Low Deficiency

It has long been conjectured that the total chromatic number. chi '' (K) of the complete p-partite graph K = K(r(1),..., r(p)) is Delta (K)+ 1 if and only if both K not equal K-r,K- r and | V(K)| = 0 (mod 2) implies that Sigma(v is an element of V)(K)( (K) -d(K) (v)) is at least the number of parts of odd size. It is known that. chi '' (K) <=Delta (K)+ 2. In this paper, a new approach is introduced to attack the conjecture that makes use of amalgamations of graphs. The power of this approach is demonstrated by settling the conjecture for all complete 5-partite graphs.