Boxicity and cubicity of a subclass of divisor graphs and power graphs of cyclic groups

Abstract

The boxicity (respectively, cubicity) of an undirected graph Γ is the smallest non-negative integer k such that Γ can be represented as the intersection graph of axis-parallel rectangular boxes (respectively, unit cubes) in $R^k$. An undirected graph is classified as a comparability graph if it is isomorphic to the comparability graph of some partial order. In this paper, we initiate the study of boxicity and cubicity for two subclasses of comparability graphs - divisor graphs and power graphs.

Divisor graphs, an important family of comparability graphs, arise from a number-theoretically defined poset, namely the divisibility poset. We consider one of the most popular subclasses of divisor graphs, denoted by $D\left(n\right)$, where the vertex set is the set of positive divisors of a natural number n, and two vertices a and b are adjacent if and only if a divides b or b divides a. We derive estimates, tight up to a factor of 2, for the boxicity and cubicity of $D\left(n\right)$.

Power graphs are a special class of algebraically defined comparability graphs. The power graph of a group is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We show that studying the boxicity (respectively, cubicity) of $D\left(n\right)$ is sufficient to study the boxicity (respectively, cubicity) of the power graph of the cyclic group of order n. Thus, as a corollary of our first result, we derive similar estimates for the boxicity and cubicity power graphs of cyclic groups.