More Results on Italian Domination in C n ? C m

Hong Gao, Penghui Wang, Enmao Liu and Yuansheng Yang
Additional contact information
Hong Gao: College of Science, Dalian Maritime University, Dalian 116026, China
Penghui Wang: College of Science, Dalian Maritime University, Dalian 116026, China
Enmao Liu: College of Science, Dalian Maritime University, Dalian 116026, China
Yuansheng Yang: School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China

Mathematics, 2020, vol. 8, issue 4, 1-10

Abstract: 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). St?pie? 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.

Keywords: rainbow domination; Roman domination; Italian domination; Cartesian product graph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/4/465/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/4/465/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:4:p:465-:d:337107

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().