On total chromatic number of complete multipartite graphs

Abstract

In 1995, Hoffman and Rodger conjectured that the total chromatic number ${\chi }^{\prime \prime }\left(K\right)$ of the complete $p$-partite graph $K=K(r_1,\dots ,r_p)$ is $\Delta \left(K\right)+1$ if and only if $K\ne K_{r,r}$ and if $K$ has an even number of vertices then $def\left(K\right)={\Sigma }_{v\in V\left(K\right)}(\Delta \left(K\right)-d_K\left(v\right))$ is at least the number of parts of odd size. The conjecture is known to be true when $K$ has odd number of vertices. When $K$ is even, the problem is quite difficult and is still open with little progress being made. The problem was settled for complete 3-partite graphs by Chew and Yap in 1992, and for complete 4-partite graphs by Dong and Yap in 2000; the difficulty rises manifold with the increase in the number of parts. In 2014, Dalal and Rodger (Graphs and Combinatorics (2015), 1–15) introduced an approach using amalgamations to attack the conjecture and demonstrated its power by settling the problem for complete 5-partite graphs. Their approach required coloring of all the vertices in each part with the same color. However, if the conjecture is true, then for each $3\le k\in N$, there are complete $2k$-partite graphs $K$ for which any total coloring of $K$ in which all the vertices in each part are colored the same would require at least $\Delta \left(K\right)+2$ colors, although ${\chi }^{\prime \prime }\left(K\right)=\Delta \left(K\right)+1$. In this paper, we provide a generalized technique that allows the vertices in the same part to have different colors by adapting a result of Bahmanian and Rodger (J. Graph Theory (2012), 297–317) on graph amalgamations. Using our technique, we solve the classification problem for all complete 6-partite graphs.