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 Brešar et al. (Discrete Appl Math 155:2394–2400, 2007). Finally, Vizing’s conjecture on Italian domination in \(P_n\Box P_m\) is checked.

在图形\（G =（V，E）\）中，每个顶点\（v \ in V \）被分配0、1或2，以便每个被分配0的顶点与至少一个被分配2个或两个顶点的顶点相邻。分配1.这样的分配称为G的意大利支配功能（IDF）。IDF f的权重为\（w（f）= \ sum _ {v \ in V} f（v）\）。G的意大利支配数为\（\ gamma _ {I}（G）= \ min _ {f} w（f）\）。在本文中，我们研究了路径的笛卡尔积\（P_n \ Box P_m \）的意大利支配数。我们获得\（\ gamma _ {I}（P_n \ Box P_2）\）和\（\ gamma _ {I}（P_n \ Box P_3）\）的精确值。此外，我们为\（m \ ge 4 \）表示\（\ gamma _ {I}（P_n \ Box P_m）\）的边界，即\（\ frac {mn} {3} + \ frac {m + n-4} {9} \ le \ gamma _ {I}（P_ {n} \ Box P_ {m}）\ le \ frac {mn + 2m + 2n-8} {3} \），其中下限由于一般下限是Chellali等人提出的\（\ frac {mn} {3} \），因此得到了改进。（离散应用数学204：22-28，2016年）。根据本文的结果以及现有结果，我们给出\（P_n \ Box P_2 \）和\（P_n \ Box P_3 \）是\（\ gamma _ {I} = \ gamma _ {r2} \）其中\（\ gamma _ {r2} \）是2彩虹的统治数字。这可以部分解决Brešar等人提出的开放问题。（离散Appl数学155：2394-2400，2007年）。最后，检查了维辛关于\（P_n \ Box P_m \）中意大利统治地位的猜想。