  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}$.
