Rainbow Domination in Cartesian Product of Paths and Cycles

Abstract

Let [Formula: see text] be a graph and [Formula: see text] be an integer representing [Formula: see text] colors. There is a function [Formula: see text] from [Formula: see text] to the power set of [Formula: see text] colors satisfying every vertex [Formula: see text] assigned [Formula: see text] under [Formula: see text] in its neighborhood has all the colors, then [Formula: see text] is called a [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) on [Formula: see text]. The weight of [Formula: see text] is the sum of the number of colors on each vertex all over the graph. The [Formula: see text]-rainbow domination number of [Formula: see text] is the minimum weight of [Formula: see text]RDFs on [Formula: see text], denoted by [Formula: see text]. The aim of this paper is to investigate the [Formula: see text]-rainbow ([Formula: see text]) domination number of the Cartesian product of paths [Formula: see text] and the Cartesian product of paths and cycles [Formula: see text]. For [Formula: see text], we determine the value [Formula: see text] and present [Formula: see text] for [Formula: see text]. For [Formula: see text], we determine the values of [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text] for [Formula: see text] or [Formula: see text].