Boxicity and Cubicity of Divisor Graphs and Power Graphs
L. Sunil Chandran, Jinia Ghosh
TL;DR
The paper analyzes boxicity and cubicity for two natural classes of comparability graphs—divisor graphs $D(n)$ and power graphs (notably for cyclic groups)—by linking them to transitive closures of Cartesian products of complete graphs. It shows $D(n)$ is isomorphic to $TC\big(a_1,\dots,a_s\big)$ and that for TC-based graphs one can derive tight-ish bounds: $box(TCC(m_1,\dots,m_d))\le cub(TCC(\cdots))\le m_1+\cdots+m_{d-1}$ with a general lower bound given by a sum of terms in the $m_i$. For squarefree $n$ (or more generally for cyclic groups of order $n$), this yields $box(D(n))\ge s/2$ (where $s$ is the number of distinct primes in $n$) and connects to known poset-dimension results. The work also shows $R(G)$, the reduced power graph of a finite cyclic group $G$, is isomorphic to $D(n)$, tying divisor graphs directly to power graphs; the derived bounds improve on standard estimates and show the boxicity/cubicity of these algebraically defined comparability graphs are governed by the prime-factor structure of $n$.
Abstract
The \textit{boxicity} (\textit{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 (unit cubes) in $\mathbb{R}^k$. An undirected graph is classified as a \textit{comparability graph} if it is isomorphic to the comparability graph of some partial order. This paper studies boxicity and cubicity for subclasses of comparability graphs. We initiate the study of boxicity and cubicity of a special class of algebraically defined comparability graphs, namely the \textit{power 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 analyse the case when the underlying groups of power graphs are cyclic. Another important family of comparability graphs is \textit{divisor graphs}, which arises from a number-theoretically defined poset, namely the \textit{divisibility poset}. We consider a subclass of divisor graphs, denoted by $D(n)$, where the vertex set is the set of positive divisors of a natural number $n$. We first show that to study the boxicity (cubicity) of the power graph of the cyclic group of order $n$, it is sufficient to study the boxicity (cubicity) of $D(n)$. We derive estimates, tight up to a factor of $2$, for the boxicity and cubicity of $D(n)$. The exact estimates hold good for power graphs of cyclic groups.