Rocky Mountain Journal of Mathematics

Dominating sets in intersection graphs of finite groups

Selcuk Kayacan

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Abstract

Let $G$ be a group. The intersection graph $\Gamma (G)$ of $G$ is an undirected graph without loops and multiple edges, defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$, where $1$ denotes the trivial subgroup of $G$. In this paper, we study the dominating sets in intersection graphs of finite groups. We classify abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related to Burnside rings. We introduce the notion of an intersection graph of a $G$-set (somewhat generalizing the ordinary definition of an intersection graph of a group) and establish a general upper bound for the domination number of $\Gamma (G)$ in terms of subgroups satisfying a certain property in the Burnside ring. The intersection graph of $G$ is the $1$-skeleton of the simplicial complex. We name this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that, if the domination number of $\Gamma (G)$ is 1, then the intersection complex of $G$ is contractible.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2311-2335.

Dates
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1544756811

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2311

Mathematical Reviews number (MathSciNet)
MR3892134

Zentralblatt MATH identifier
06999264

Subjects
Primary: 20D99: None of the above, but in this section
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C69: Dominating sets, independent sets, cliques 20C05: Group rings of finite groups and their modules [See also 16S34] 55U10: Simplicial sets and complexes

Keywords
Finite groups subgroup intersection graph dominating sets domination number Burnside ring order complex

Citation

Kayacan, Selcuk. Dominating sets in intersection graphs of finite groups. Rocky Mountain J. Math. 48 (2018), no. 7, 2311--2335. doi:10.1216/RMJ-2018-48-7-2311. https://projecteuclid.org/euclid.rmjm/1544756811


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