Total $k$-domination in Cartesian product of complete graphs

TL;DR

This paper investigates the total k-domination number of Cartesian products of complete graphs, establishing new bounds, asymptotic behaviors, and exact formulas for specific cases, advancing understanding of domination parameters in graph products.

Contribution

It provides new bounds, asymptotic results, and exact formulas for the total k-domination number in Cartesian products of complete graphs, including special cases and improvements for k=3.

Findings

Derived new lower and upper bounds for the total k-domination number.

Established asymptotic behavior and limit infimum for large graphs.

Obtained exact formulas for the case k=2 and improved bounds for k=3.

Abstract

Let $G=(V,E)$ be a finite undirected graph. A set $S$ of vertices in $V$ is said to be total $k$-dominating if every vertex in $V$ is adjacent to at least $k$ vertices in $S$. The total $k$-domination number, $\gamma_{kt}(G)$, is the minimum cardinality of a total $k$-dominating set in $G$. In this work we study the total $k$-domination number of Cartesian product of two complete graphs which is a lower bound of the total $k$-domination number of Cartesian product of two graphs. We obtain new lower and upper bounds for the total $k$-domination number of Cartesian product of two complete graphs. Some asymptotic behaviors are obtained as a consequence of the bounds we found. In particular, we obtain that $\displaystyle\liminf_{n\to\infty}\frac{\gamma_{kt}(G\Box H)}{n}\leq 2\,\left(\left\lceil\frac{k}{2}\right\rceil^{-1}+\left\lfloor\frac{k+4}{2}\right\rfloor^{-1}\right)^{-1}$ for graphs $G,H$ with order at least $n$. We also prove that the equality is attained if and only if $k$ is even. The equality holds when $G,H$ are both isomorphic to the complete graph, $K_n$, with $n$ vertices. Furthermore, we obtain closed formulas for the total $2$-domination number of Cartesian product of two complete graphs of whatever order. Besides, we prove that, for $k=3$, the inequality above is improvable to $\displaystyle\liminf_{n\to\infty} \gamma_{3t}(K_n\Box K_n)/n \leq 11/5$.