The Total Chromatic Number of Complete Multipartite Graphs with Low Deficiency

Abstract

It has long been conjectured that the total chromatic number $\chi ^{\prime \prime }(K)$ of the complete $p$-partite graph $K = K(r_1, \dots , r_p)$ is $\Delta (K) + 1$ if and only if both $K \ne K_{r,r}$ and $|V(K)| \equiv$0 (mod 2) implies that $\Sigma _{v \in V(K)}(\Delta (K) - d_K(v))$ is at least the number of parts of odd size. It is known that $\chi ^{\prime \prime }(K) \le \Delta (K) + 2$. 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.