Italian domination in the Cartesian product of paths

In a graph G = (V, E), each vertex v is an element of 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) = Sigma(v is an element of V) f (v). The Italian domination number of G is gamma(I) (G) = min(f) w(f). In this paper, we investigate the Italian domination number of the Cartesian product of paths, P-n square P-m. We obtain the exact values of gamma(I) (P-n square P-2) and gamma(I) (P-n square P-3). Also, we present a bound of gamma(I) (P-n square P-m) for m >= 4, that is mn/3 + m+n-4/9 <= gamma(I) (P-n square P-m) <= mn+2m+2n-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 square P-2 and P-n square P-3 are examples for which gamma(I) = gamma(r2) where gamma(r2) is the 2-rainbow domination number. This can partially solve the open problem presented by Bresar et al. (Discrete Appl Math 155:2394-2400, 2007). Finally, Vizing's conjecture on Italian domination in P-n square P-m is checked.