Italian domination in the Cartesian product of paths

In a graph $$G=(V,E)$$ G = ( V , E ) , each vertex $$v\in V$$ 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 $$w(f)=\sum _{v\in V}f(v)$$ w ( f ) = ∑ v ∈ V f ( v ) . The Italian domination number of G is $$\gamma _{I}(G)=\min _{f} w(f)$$ γ I ( G ) = min f w ( f ) . In this paper, we investigate the Italian domination number of the Cartesian product of paths, $$P_n\Box P_m$$ P n □ P m . We obtain the exact values of $$\gamma _{I}(P_n\Box P_2)$$ γ I ( P n □ P 2 ) and $$\gamma _{I}(P_n\Box P_3)$$ γ I ( P n □ P 3 ) . Also, we present a bound of $$\gamma _{I}(P_n\Box P_m)$$ γ I ( P n □ P m ) for $$m\ge 4$$ m ≥ 4 , that is $$\frac{mn}{3}+\frac{m+n-4}{9}\le \gamma _{I}(P_{n}\Box P_{m})\le \frac{mn+2m+2n-8}{3}$$ 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 $$\frac{mn}{3}$$ 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 $$P_n\Box P_2$$ P n □ P 2 and $$P_n\Box P_3$$ P n □ P 3 are examples for which $$\gamma _{I}=\gamma _{r2}$$ γ I = γ r 2 where $$\gamma _{r2}$$ 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 $$P_n\Box P_m$$ P n □ P m is checked.