Total colorings of equibipartite graphs Original Research Article

The total chromatic number χτ(G) of a graph G is the least number of colors needed to color the vertices and the edges of G such that no adjacent or incident pair of elements receive the same color. A simple graph G is called type 1 if χτ(G) = Δ(G) + 1, where Δ(G) is the maximum degree of G. In this paper we prove the following conjecture of Chen et al.: An (n − 2)-regular equibipartite graph Kn,n − E(J) is type 1 if and only if J contains a 4-cycle.