Italian domination in the Cartesian product of paths

In a graph G = ( V , E ) , each vertex v ∈ V is assigned 0, 1 or 2 such that each vertex assigned 0 is adjacent to at least one vertex assigned 2 or two vertices assigned 1. Such an assignment is called an Italian dominating function (IDF) of G . The weight of an IDF f is w ( f ) = ∑ v ∈ V f ( v ) . The Italian domination number of G is γ I ( G ) = min f w ( f ) . In this paper, we investigate the Italian domination number of the Cartesian product of paths, P n □ P m . We obtain the exact values of γ I ( P n □ P 2 ) and γ I ( P n □ P 3 ) . Also, we present a bound of γ I ( P n □ P m ) for m ≥ 4 , that is mn 3 + m + n - 4 9 ≤ γ I ( P n □ P m ) ≤ m n + 2 m + 2 n - 8 3 where the lower bound is improved since the general lower bound is mn 3 presented by Chellali et al. (Discrete Appl Math 204:22–28, 2016). By the results of this paper, together with existing results, we give P n □ P 2 and P n □ P 3 are examples for which γ I = γ r 2 where γ r 2 is the 2-rainbow domination number. This can partially solve the open problem presented by Brešar et al. (Discrete Appl Math 155:2394–2400, 2007). Finally, Vizing’s conjecture on Italian domination in P n □ P m is checked.