Italian domination in the Cartesian product of paths

Abstract

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.