k-Tuple total domination and Mycieleskian graphs

Let $k$ be a positive integer. A subset $S$ of $V(G)$ in a graph $G$ is a $k$-tuple total dominating set of $G$ if every vertex of $G$ has at least $k$ neighbors in $S$. The $k$-tuple total domination number $gamma _{times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$. If$V(G)=V^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $E(G)=E_{0}$, then for any integer $mgeq 1$ the $m$-emph{Mycieleskian} $mu _{m}(G)$ of $G$ is the graph with vertex set $V^{0}cup V^{1}cup V^{2}cup cdots cup V^{m}cup {u}$, where $V^{i}={v_{j}^{i}mid v_{j}^{0}in V^{0}}$ is the $i$-th distinct copy of $V^{0}$, for $% i=1,2,ldots ,m$, and edge set $E_{0}cup left( bigcup _{i=0}^{m-1}{v_{j}^{i}v_{j^{prime }}^{i+1}mid v_{j}^{0}v_{j^{prime }}^{0}in E_{0}}right) cup {v_{j}^{m}umid v_{j}^{m}in V^{m}}$. In this paper for a given graph $G$ with minimum degree at least $k$, we find some sharp lower and upper bounds on the $k$-tuple total domination number of the $m$-Mycieleskian graph $mu _{m}(G)$ of $G$ in terms on $k$ and $gamma_{times k,t}(G)$. Specially we give the sharp bounds $gamma _{times k,t}(G)+1$ and $gamma _{times k,t}(G)+k$ for $gamma_{times k,t}(mu _1(G))$, and characterize graphs with $gamma_{times k,t}(mu _1(G))=gamma _{times k,t}(G)+1$.