Signed total domination on Kronecker products of two complete graphs (Q2848726)

The Kronecker product \(G_1\otimes G_2\) of graphs \(G_1, G_2\) has vertex set \(V(G_1)\times V(G_2)\), and edge set \(\{(u_1,v_1)(u_2,v_2):u_1u_2\in E(G_1)\text{ and }v_1v_2\in E(G_2)\}\). The neighbourhood \(N(v)\) of a vertex \(v\in V(G)\) is \(\{u\in V(G):uv\in E(G)\}\). A total dominating function (respectively minus total dominating function, signed total dominating function) of \(G\) is a function \(f:V(G)\to\{0,1\}\) (respectively \(f:V(G)\to\{-1,0,1\}\), \(f:V(G)\to\{-1,1\}\)) where \(\sum\limits_{u\in N(v)}f(u)\geq1\) for every vertex \(v\in V(G)\); the \textit{weight} \(w(f)\) is \(\sum\limits_{v\in V}f(v)\); for a subset \(S\subseteq V\), \(f(S)=\sum\limits_{v\in S}f(v)\). The total domination number \(\gamma_t(G)\) (respectively minus total domination number \(\gamma_t^{-}(G)\), signed total domination number \(\gamma_{st}(G)\)) is the minimum weight among all total dominating functions (respectively minus total dominating functions, signed total dominating functions).NEWLINENEWLINENEWLINE{Theorem 2.} Let \(m, n\geq2\).NEWLINENEWLINE (1) When both \(m\) and \(n\) are even, \(\gamma_{st}(K_m\otimes K_n)=4\).NEWLINENEWLINE (2) When both \(m\) and \(n\) are odd, \(\gamma_{st}(K_m\otimes K_n)\)=\(\begin{cases}7,&\text{if }\min\{m,n\}=3\)