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

Let $G=(V,E)$ be a simple undirected graph. $G$ is a circulant graph defined on $V={\mathbb{Z}}_n$ with difference set $D\subseteq {1,2,\ldots,\lfloor\frac{n}{2}\rfloor}$providedtwovertices$i$and$j$in$$\mathbb{Z}$n$areadjacentifandonlyif$\min{|i-j|, n-|i-j|}\in D$. For convenience, we use $G(n;D)$ to denote such a circulant graph. A function $f:V(G)\to \mathbb{N}\cup \{0\}$ is an integer $\{k\}$-domination function if for each $v\in V(G)$, $\sum{u\in N_G[v]}f(u)\geq k.$Byconsideringall${k}$-dominationfunctions$f$, 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 $D=\{1,2,\dots ,t\}$, $1\le t\le \frac{n-1}{2}$, then the integer $\{k\}$-domination number of $G(n;D)$ is $\lceil \frac{kn}{2t+1}\rceil$.