Total coloring of circulant graphs $C_{n}(1, 4)$

Total coloring of circulant graphs has attracted much attention in recent years. Studies on the total chromatic numbers of them, in particular 4-regular circulant graphs, have thrown up a number of interesting results. However, as a challenging issue, the total chromatic numbers of 4-regular circulant graphs $C_{n}(1, 4)$ remain an open question even after many efforts. In this paper, we solve this question by completely determining total chromatic numbers of $C_{n}(1, 4)$ for all $n\geq 9$.