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# More Results on Italian Domination in Cn□Cm

by 1,*, 1 and
1
College of Science, Dalian Maritime University, Dalian 116026, China
2
School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 465; https://doi.org/10.3390/math8040465 (registering DOI)
Received: 25 February 2020 / Revised: 21 March 2020 / Accepted: 22 March 2020 / Published: 26 March 2020
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 $γ I$ . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained $γ I ( C n □ C 3 )$ and $γ I ( C n □ C 4 )$ (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75–85). Stȩpień et al. obtained $γ I ( C n □ C 5 ) = 2 n$ (2-Rainbow domination number of $C n □ C 5$ , Discret. Appl. Math. 170 (2014): 113–116). In this paper, we study the Italian domination number of the Cartesian products of cycles $C n □ C m$ for $m ≥ 6$ . For $n ≡ 0 ( mod 3 )$ , $m ≡ 0 ( mod 3 )$ , we obtain $γ I ( C n □ C m ) = m n 3$ . For other $C n □ C m$ , we present a bound of $γ I ( C n □ C m )$ . Since for $n = 6 k$ , $m = 3 l$ or $n = 3 k$ , $m = 6 l$ $( k , l ≥ 1 )$ , $γ r 2 ( C n □ C m ) = m n 3$ , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668–674), it follows in this case that $C n □ C m$ is an example of a graph class for which $γ I = γ r 2$ , 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.
MDPI and ACS Style

Gao, H.; Wang, P.; Liu, E.; Yang, Y. More Results on Italian Domination in CnCm. Mathematics 2020, 8, 465.