Total-colorings of complete multipartite graphs using amalgamations

This paper makes progress towards settling the long-standing conjecture that the total chromatic number χ″χ″ of the complete pp-partite graph K=K(r1,…,rp)K=K(r1,…,rp) is Δ(K)+1Δ(K)+1 if and only if K≠Kr,rK≠Kr,r and if KK has an even number of vertices then Σv∈V(K)(Δ(K)−dK(v))Σv∈V(K)(Δ(K)−dK(v)) is at least the number of parts of odd size. Graphs of even order that are fairly close to being regular are the ones for which χ″(K)χ″(K) remains in doubt. In this paper we show that KK is of Type 1 if |V(K)||V(K)| is even and r2