More Results on Italian Domination in Cn□Cm

Mathematics

Penghui Wang¹  Enmao Liu¹  Hong Gao¹  Yuansheng Yang² 

[1] College of Science, Dalian Maritime University, Dalian 116026, China;School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China;

关键词: rainbow domination;    roman domination;    italian domination;    cartesian product graph;   

DOI  :  10.3390/math8040465

来源: DOAJ

【 摘 要 】

Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In such an assignment, the minimum number of troops is the Italian domination number of the empire/graph is denoted as ${\gamma }_I$ . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained ${\gamma }_I(C_n\square C_3)$ and ${\gamma }_I(C_n\square C_4)$ (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75−85). Stȩpień et al. obtained ${\gamma }_I(C_n\square C_5)=2n$ (2-Rainbow domination number of $C_n\square C_5$ , Discret. Appl. Math. 170 (2014): 113−116). In this paper, we study the Italian domination number of the Cartesian products of cycles $C_n\square C_m$ for $m\ge 6$ . For $n\equiv 0(mod3)$ , $m\equiv 0(mod3)$ , we obtain ${\gamma }_I(C_n\square C_m)=\frac{mn}{3}$ . For other $C_n\square C_m$ , we present a bound of ${\gamma }_I(C_n\square C_m)$ . Since for $n=6k$ , $m=3l$ or $n=3k$ , $m=6l$ $(k,l\ge 1)$ , ${\gamma }_{r2}(C_n\square C_m)=\frac{mn}{3}$ , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668−674), it follows in this case that $C_n\square C_m$ is an example of a graph class for which ${\gamma }_I={\gamma }_{r2}$ , which can partially answer the question presented by Brešar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394−2400.

【 授权许可】

Unknown