Total colouring of some cartesian and direct product graphs
Kyle MacKeigan, Jeannette Janssen · Contributions to Discrete Mathematics · 2020
A graph is $k$-total colourable if there is an assignment of $k$ different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product graphs are determined. In particular, a sufficient condition is given for direct products of bipartite graphs to have total chromatic number equal to its maximum degree plus one. Partial results towards the total chromatic number of $K_n\times K_m$ are also established.