The 2-rainbow domination number of Cartesian product of cycles

TL;DR

This paper investigates the 2-rainbow domination number of the Cartesian product of cycles, providing exact values for some families and bounds for others, advancing understanding of rainbow domination in graph products.

Contribution

It introduces new bounds and exact values for the 2-rainbow domination number of Cartesian products of cycles, a novel focus in graph theory.

Findings

Exact values for specific cycle families.

Lower and upper bounds for general cases.

Enhanced understanding of rainbow domination in graph products.

Abstract

A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $ \{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\emptyset $ we have $\bigcup\nolimits_{u\in N(v)}f(u)=\{1,2,...,k\}$. The weight $w(f)$ of a $k$RDF $f$ is defined as $w(f)=\sum_{v\in V(G)}\left\vert f(v)\right\vert $. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, which is denoted by $\gamma_{rk}(G)$. In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases.