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 $V=\{1,\dots ,n\}$ 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 $n\ge 6$, $gcd(a,n)=1$ and $a^{-1}b\equiv 3(\mathrm{mod}n)$.

Accepted for publishing at Utilitas Mathematica, April 23, 2019