Total domination in cubic Knödel graphs

Abstract: A subset $D$ of vertices of a graph $G$ is a \textit{dominating set} if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The \textit{dominating number}, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. A set $D\subseteq V(G)$ is a \textit{total dominating set} if for each $u\in V(G)$, $u$ is adjacent to some vertex $v\in D$. the The \textit{total dominating number}, $\gamma_t(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\ge2$ and $1\le\Delta\le\lfloor\log_2n\rfloor$, a \textit{Kn\"odel graph} $W_{\Delta,n}$ is a $\Delta$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for $i=1,2$ and $0\le j\le n/2-1$, where for every $j$,$0\le j\le n/2-1$,there is an edge between vertex $(1,j)$ and every vertex $(2,j+2^k-1 \text{(mod(n/2)})$, for $k=0,1,\cdots,\Delta-1$. In this paper, we determine the total domination number in $3$-regular Kn\"odel graphs $W_{3,n}$.