Italian domination in the Cartesian product of paths

In a graph $$G=(V,E)$$ , each vertex $$v\in V$$ 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 $$w(f)=\sum _{v\in V}f(v)$$ . 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, $$P_n\Box P_m$$ . We obtain the exact values of $$\gamma _{I}(P_n\Box P_2)$$ and $$\gamma _{I}(P_n\Box P_3)$$ . Also, we present a bound of $$\gamma _{I}(P_n\Box P_m)$$ for $$m\ge 4$$ , that is $$\frac{mn}{3}+\frac{m+n-4}{9}\le \gamma _{I}(P_{n}\Box P_{m})\le \frac{mn+2m+2n-8}{3}$$ where the lower bound is improved since the general lower bound is $$\frac{mn}{3}$$ presented by Chellali et al. (Discrete Appl Math 204:22–28, 2016). By the results of this paper, together with existing results, we give $$P_n\Box P_2$$ and $$P_n\Box P_3$$ 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 $$P_n\Box P_m$$ is checked.