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

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.