Total Colourings of Direct Product Graphs

TL;DR

This paper investigates the total chromatic number of direct product graphs, providing new conditions for bipartite graphs and partial results for complete graphs, advancing understanding of graph coloring complexities.

Contribution

It introduces a sufficient condition for direct products of bipartite graphs to have total chromatic number equal to maximum degree plus one and offers partial results for complete graphs.

Findings

Sufficient condition established for bipartite graph products

Total chromatic number determined for some direct product graphs

Partial results obtained for complete graph products

Abstract

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 the direct product of complete graphs are also established.