# On the integer {k}-domination number of circulant graphs

## TL;DR

This paper determines the exact integer {k}-domination number for a class of circulant graphs with specific difference sets, providing a precise formula based on graph parameters.

## Contribution

It introduces a closed-form expression for the {k}-domination number of circulant graphs with difference set D={1,2,...,t}, extending understanding of domination in these graphs.

## Key findings

- Derived the {k}-domination number formula for D={1,2,...,t}
- Established the exact value as a ceiling function involving n, k, and t
- Enhanced the theoretical framework for domination in circulant graphs.

## Abstract

Let $G=(V,E)$ be a simple undirected graph. $G$ is a circulant graph defined on $V=\mathbb{Z}_n$ with difference set $D\subseteq \{1,2,\ldots,\lfloor\frac{n}{2}\rfloor\}$ provided two vertices $i$ and $j$ in $\mathbb{Z}_n$ are adjacent if and only if $\min\{|i-j|, n-|i-j|\}\in D$. For convenience, we use $G(n;D)$ to denote such a circulant graph. A function $f:V(G)\rightarrow\mathbb{N}\cup\{0\}$ is an integer $\{k\}$-domination function if for each $v\in V(G)$, $\sum_{u\in N_G[v]}f(u)\geq k.$ By considering all $\{k\}$-domination functions $f$, the minimum value of $\sum_{v\in V(G)}f(v)$ is the $\{k\}$-domination number of $G$, denoted by $\gamma_k(G)$. In this paper, we prove that if $D=\{1,2,\ldots,t\}$, $1\leq t\leq \frac{n-1}{2}$, then the integer $\{k\}$-domination number of $G(n;D)$ is $\lceil\frac{kn}{2t+1}\rceil$.