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

TL;DR

This paper determines the total dominator chromatic number of specific circulant graphs with two connection steps, under certain gcd and modular conditions, expanding understanding of graph coloring properties in circulant graphs.

Contribution

It provides explicit calculations of the total dominator chromatic number for circulant graphs with two connection steps under particular gcd and modular constraints.

Findings

Calculated total dominator chromatic number for $C_n(oxed{a,b})$ with given conditions.

Extended existing graph coloring results to a new class of circulant graphs.

Identified specific conditions where the chromatic number can be explicitly determined.

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}$.