Total Colorings of Some Classes of FourRegular Circulant Graphs

TL;DR

This paper determines the total chromatic number for specific classes of four-regular circulant graphs, contributing to the understanding of total colorings in this graph class.

Contribution

It provides the total chromatic number for certain classes of four-regular circulant graphs, advancing knowledge on the Total Coloring Conjecture for these graphs.

Findings

Total chromatic number computed for some four-regular circulant graphs

Supports the Total Coloring Conjecture in these cases

Provides new results for graph coloring theory

Abstract

The total chromatic number, $\chi''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph, $\chi''(G) \leq \Delta(G)+2 $, where $\Delta(G)$ represents the maximum degree of $G$. In this paper we obtained the total chromatic number for some classes of four regular circulant graphs.