k -Rainbow Domination Number of P 3 □ P n

摘要

Let k be a positive integer, and set [ k ] : = { 1 , 2 , … , k } . For a graph G , a k-rainbow dominating function (or kRDF ) of G is a mapping f : V ( G ) → 2 [ k ] in such a way that, for any vertex v ∈ V ( G ) with the empty set under f , the condition ? u ∈ N G ( v ) f ( u ) = [ k ] always holds, where N G ( v ) is the open neighborhood of v . The weight of kRDF f of G is the summation of values of all vertices under f . The k-rainbow domination number of G , denoted by γ r k ( G ) , is the minimum weight of a kRDF of G . In this paper, we obtain the k-rainbow domination number of grid P 3 □ P n for k ∈ { 2 , 3 , 4 } .