Boxicity of Power Graphs of Finite Nilpotent Groups

Abstract

The \textit{boxicity} of an undirected graph $Γ$, denoted as $\operatorname{box}(Γ)$, is the smallest non-negative integer $d$ such that $Γ$ can be represented as the intersection graph of axis-parallel rectangular boxes in $\mathbb{R}^d$. The \emph{power graph} of a group $G$, denoted as $\operatorname{Pow}(G)$, 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 determine the exact boxicity of power graphs of finite nilpotent groups. For a cyclic group of order $n=\prod_{i=1}^{r}p_i^{e_i}>1$, where the $p_i$ are distinct primes, we prove that its power graph has boxicity $\sum_{i=1}^{r}e_i-\max_i e_i$. For a finite nilpotent group $G$, let the number of noncyclic odd Sylow subgroups be denoted by $t(G)$. If $G$ has odd order or if $G$ has a cyclic Sylow $2$-subgroup, then $\operatorname{box}(\operatorname{Pow}(G))=\operatorname{box}(\operatorname{Pow}(\mathbb{Z}_{\operatorname{exp}(G)}))+t(G)$, where $\operatorname{exp}(G)$ denotes the \emph{exponent} of a group $G$. When the Sylow $2$-subgroup is noncyclic, and the odd part is nontrivial, the boxicity equals this value or one more, and we characterise precisely when each of these values is attained. The boxicity of power graphs of arbitrary finite groups and the exact boxicity of the comparability graph of the Boolean lattice were posed as open questions in Discrete Mathematics, 349(7):115017, 2026. As a consequence of our results, we answer the latter question completely and the former question partially, by computing the exact boxicity of the power graphs of nilpotent groups.