Total colouring of circulant graphs $C_{n}(1, 3)$

TL;DR

This paper determines the total chromatic numbers of circulant graphs $C_{n}(1, 3)$, revealing specific values for certain n and establishing the exact total chromatic number for all others.

Contribution

It provides the first complete determination of the total chromatic numbers for the class of circulant graphs $C_{n}(1, 3)$, solving an open problem.

Findings

Total chromatic number is 6 for n=7,8,12,13,17.

Total chromatic number is 5 for all other n.

The results resolve an open question in circulant graph coloring.

Abstract

Total colouring of 4-regular circulant graphs is an interesting but challenging topic, and has attracted much attention. However, it still remains an open question to determine the total chromatic numbers of $C_{n}(1, 3)$, a subclass of 4-regular circulant graphs, even after many efforts. In this paper, we investigate the total colouring of these graphs and determine their total chromatic numbers. Our results show that the total chromatic numbers of $C_{n}(1, 3)$ are 6 for $n=7,8,12,13,17$, and 5 for all others.