The Total Chromatic Number of Some Bipartite Graphs

Abstract

The total chromatic number ${\chi }_{\tau }(G)$ is the least number of colours needed to colour the vertices and edges of a graph $G$ such that no incident or adjacent elements (vertices or edges) receive the same colour. This work determines the total chromatic number of grids, particular cases of partial grids, near-ladders, and of $k$-dimensional cubes.