On the integer {k}-domination number of 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 $\lceil\frac{kn}{2t+1}\rceil$.