On a Conjecture About Signed Domination in the Cartesian Product of Two Directed Cycles

Abstract

Let D be a finite and simple digraph with vertex set V(D). A signed dominating function (SDF) of D is a function $$f:V(D)\longrightarrow \{-1,1\}$$ such that $$f(N^{-}[v])=\sum _{x\in N^{-}[v]}f(x)\ge 1$$ for every $$v\in V(D)$$ , where $$N^{-}[v]$$ consists of v and all vertices of D from which arcs go into v. The weight of an SDF is the sum of its function values over all vertices, and the minimum weight of an SDF of G is the signed domination number $$\gamma _{s}(D).$$ In this paper, we investigate the signed domination number of the Cartesian product of two directed cycles by showing that $$\gamma _{s}(C_{m}\Box C_{n})=\lceil \frac{m}{3}\rceil n$$ if $$n\equiv 0\pmod {2m}$$ or $$n\ge m$$ and $$m\equiv 1\pmod 3,$$ answering a conjecture posed in Shaheen (J Progress Res Math 6(2):770–777, 2016). Moreover, the exact value of $$\gamma _{s}(C_{8}\Box C_{n})$$ is also provided.