More Results on Italian Domination in Cn□Cm

Autor: Enmao Liu, Hong Gao, Penghui Wang, Yuansheng Yang
Jazyk: angličtina
Rok vydání: 2020
Předmět:
Zdroj: Mathematics
Volume 8
Issue 4
Mathematics, Vol 8, Iss 4, p 465 (2020)
ISSN: 2227-7390
DOI: 10.3390/math8040465
Popis: 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 □ C 3 ) and &gamma
I ( C n □ C 4 ) (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75&ndash
85). Stȩpień et al. obtained &gamma
I ( C n □ C 5 ) = 2 n (2-Rainbow domination number of C n □ C 5 , Discret. Appl. Math. 170 (2014): 113&ndash
116). In this paper, we study the Italian domination number of the Cartesian products of cycles C n □ C m for m &ge
6 . For n &equiv
0 ( mod 3 ) , m &equiv
0 ( mod 3 ) , we obtain &gamma
I ( C n □ C m ) = m n 3 . For other C n □ C m , we present a bound of &gamma
I ( C n □ C m ) . Since for n = 6 k , m = 3 l or n = 3 k , m = 6 l ( k , l &ge
1 ) , &gamma
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&ndash
674), it follows in this case that C n □ C m is an example of a graph class for which &gamma
I = &gamma
r 2 , which can partially answer the question presented by Bre&scaron
ar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394&ndash
2400.
Databáze: OpenAIRE
Externí odkaz:
Nepřihlášeným uživatelům se plný text nezobrazuje