A Note on Total and Paired Domination of Cartesian Product Graphs

TL;DR

This paper explores bounds on total and paired domination numbers in Cartesian product graphs, extending classical results and Vizing's conjecture to these variants and higher Cartesian powers.

Contribution

It introduces new bounds for total and paired domination numbers in Cartesian products, extending existing inequalities to multiple graph products.

Findings

Established bounds for total domination in Cartesian products.

Extended domination inequalities to n-fold Cartesian products.

Modified Clark and Suen's approach for new domination bounds.

Abstract

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in $G$. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G) \gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G) \gamma(H) \leq 2\gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.