Total dominator coloring of circulant graphs $C_n(a,b)$

Abstract: The circulant graph $C_n(S)$ with connection set $S\subseteq {1,2,\cdots,n}$ is the graph with vertex set $V={1,\ldots, n}$ and two vertices $x,y$ are adjacent if $|x-y|\in S$. In this paper, we will calculate the total dominator chromatic number of the circulant graph $C_n({a,b})$ when $n\geq 6$, $gcd(a,n)=1$ and $ a^{-1}b\equiv 3 \pmod{n}$.