Total domination in cubic Kn\"odel graphs

A subset $D$ of vertices of a graph $G$ is a $\mathit{\text{dominating}}\mathit{\text{set}}$ if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The $\mathit{\text{dominating}}\mathit{\text{number}}$, $\gamma (G)$ 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 $\mathit{\text{total}}\mathit{\text{dominating}}\mathit{\text{number}}$, ${\gamma }_t(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\ge 2$ and $1\le \mathrm{\Delta }\le \lfloor {\log }_{2}n\rfloor$, a $\mathit{\text{Kn}}\ddot{\mathit{\text{o}}}\mathit{\text{del}}\mathit{\text{graph}}$ $W_{\mathrm{\Delta },n}$ is a $\mathrm{\Delta }$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for $i=1,2$ and $0\le j\le n\mathrm{/}2-1$, where for every $j$,$0\le j\le n\mathrm{/}2-1$,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 $W_{3,n}$.