[Paper Review] Domination Parameters of the Unitary Cayley Graph of $\mathbb{Z}/n\mathbb{Z}$

Amanda Burcroff|arXiv (Cornell University)|Sep 13, 2018

Graph Labeling and Dimension Problems22 references3 citations

TL;DR

This paper investigates domination parameters in the unitary Cayley graph $X_n$ over $\mathbb{Z}/n\mathbb{Z}$, focusing on total domination, irredundance, and independence numbers. It constructs infinite families of integers $n$ with arbitrarily many prime factors such that the total domination number $\gamma_t(X_n) \leq g(n) - 2$, where $g(n)$ is the Jacobsthal function, thus answering two open questions of Defant and Iyer positively.

ABSTRACT

The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a family of infinitely many integers $n$ such that $γ_t(X_n) \leq g(n) - 2$, where $γ_t$ denotes the total domination number and $g$ denotes the Jacobsthal function. We determine the irredundance number, domination number, and lower independence number of certain direct products of complete graphs and give bounds for these parameters for any direct product of complete graphs. We provide upper bounds on the size of irredundant sets in direct products of balanced, complete multipartite graphs which are asymptotically correct for the unitary Cayley graphs of integers with a bounded smallest prime factor.

Motivation & Objective

• To resolve open questions about the existence of integers $n$ with $\gamma_t(X_n) \leq g(n)-2$ and $\gamma(X_n) \leq g(n)-2$.
• To determine domination parameters—irredundance, domination, and lower independence numbers—for direct products of complete graphs.
• To extend results on $X_n$ by analyzing its structure as a product of balanced complete multipartite graphs.
• To explore the dependence of domination parameters on the sizes of component graphs in higher-order products.
• To conjecture that $\operatorname{IR}(G) = \alpha(G)$ for products of balanced complete multipartite graphs, generalizing known results.

Proposed method

• Uses the isomorphism $X_n \cong \prod_{i=1}^t K[p_i^{\alpha_i-1}, p_i]$ via the Chinese Remainder Theorem for prime powers.
• Applies combinatorial arguments and the Pigeonhole Principle to bound the size of irredundant sets in direct products of complete multipartite graphs.
• Constructs dominating cycles of size $g(n)-2$ in $X_n$ to establish upper bounds on $\gamma_t(X_n)$.
• Analyzes private neighbors and vector projections to derive contradictions in assumed irredundant set sizes.
• Leverages known results on gcd-graphs and eigenvalue integrality to suggest extensions to broader graph families.
• Adapts techniques from Defant and Iyer to compute upper irredundance and upper domination numbers in products of balanced complete multipartite graphs.

Experimental results

Research questions

• RQ1 Do there exist infinitely many integers $n$ such that $\gamma_t(X_n) \leq g(n) - 2$?
• RQ2 Can such integers $n$ have arbitrarily many distinct prime factors?
• RQ3 Is there a single integer $n$ for which $\gamma_t(X_n) \leq g(n) - 3$?
• RQ4 For which squarefree $n$ does $\operatorname{ir}(X_n) = i(X_n)$ hold?
• RQ5 When do domination parameters of $\prod_{i=1}^t K_{n_i}$ depend on all $n_i$?

Key findings

• The paper constructs an infinite family of integers $n$ with arbitrarily many distinct prime factors such that $\gamma_t(X_n) \leq g(n) - 2$, answering two questions of Defant and Iyer affirmatively.
• It proves that $\gamma_t(X_n) \leq g(n) - 2$ holds for infinitely many $n$, with $X_n$ containing a dominating cycle of size $g(n) - 2$, which is a total dominating set.
• For direct products of complete graphs, the paper determines the irredundance, domination, and lower independence numbers in cases with at most three factors.
• It shows that $\operatorname{ir}(X_n) = i(X_n)$ when $n$ is prime, $n = 2p$, $n = 3p$ for prime $p$, or $n$ is squarefree with exactly three prime divisors.
• The paper establishes that the domination number of $K_a \times K_b \times K_c$ is independent of $a$, $b$, and $c$, but for four factors, it depends on all four sizes.
• It conjectures that $\operatorname{IR}(G) = \alpha(G)$ for any product $G = \prod_{i=1}^t K[a_i, b_i]$, extending a result on strongly perfect graphs.