Total coloring of 4-regular circulant graphs and Abelian graphs

In this paper, we study the total chromatic numbers of three families of Cayley graphs. We proved that the 4-regular circulant graph $C_{4 n}(1,4)$ for any integer $n\gt2$ and $C_{4 \lambda n}(1, n)$ for integers $\lambda$ and n, when $\lambda \geq 1$ and n is even are Type 1, which verifies the conjecture proposed by Khennoufa and Togni. In addition, we proved that the 4-regular Cayley graph $\Gamma\left(\mathbb{Z}_{2 n} \times\right.$ $\left.\mathbb{Z}_{3 m}, S\right)$ is Type 1, where $S=\left\{a^{-1}, a, a b, a^{-1} b\right\}$