On domination of Cartesian product of directed cycles

TL;DR

This paper investigates the domination number of the Cartesian product of directed cycles, providing new exact values for specific cases, improving bounds for others, and disproving a previously conjectured formula.

Contribution

It determines the domination number for cases where m ≡ 2 (mod 3), improves lower bounds for remaining cases, and refutes a conjecture for m ≡ 0 (mod 3).

Findings

Exact values for γ(C_m □ C_n) when m ≡ 2 (mod 3)

Improved lower bounds for most remaining cases

Disproof of the conjectured formula for m ≡ 0 (mod 3)

Abstract

Let $\gamma(C_m\Box C_n)$ be the domination number of the Cartesian product of directed cycles $C_m$ and $C_n$ for $m,n\geq2$. Shaheen [] and Liu and al.[ ], [ ] determined the value of $\gamma(C_m\Box C_n)$ when $m \leq 6$ and when both $m$ and $n$ $\equiv 0$ $(mod\: 3)$. In this article we give, in general, the value of $\gamma(C_m\Box C_n)$ when $m\equiv 2$ $(mod\: 3)$ and improve the known lower bound for most of the remaining cases. We also disprove the conjectured formula for the case $m$ $\equiv 0$ $(mod\: 3)$ appearing in \cite{}