$3$-tuple total domination number of rook's graphs

TL;DR

This paper derives a general formula for the 3-tuple total domination number of rook's graphs, specifically for the Cartesian product of two complete graphs, providing a constructive proof.

Contribution

It presents a new explicit formula for the 3-tuple total domination number of rook's graphs and offers a constructive proof for this formula.

Findings

Provides a formula for γ_{×3,t}(K_n □ K_m)

Constructive proof of the formula

Enhances understanding of domination in rook's graphs

Abstract

A $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$. The minimum size of a $k$TDS is called the $k$-tuple total dominating number and it is denoted by $\gamma_{\times k,t}(G)$. We give a constructive proof of a general formula for $\gamma_{\times 3, t}(K_n \Box K_m)$.