Boxicity of Power Graphs of Finite Nilpotent Groups
Abstract
The \textit{boxicity} of an undirected graph Γ, denoted as 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 ℝd. The \emph{power graph} of a group G, denoted as 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=∏i=1rpiei>1, where the pi are distinct primes, we prove that its power graph has boxicity ∑i=1rei−maxi ei. 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 box(Pow(G))=box(Pow(ℤexp(G)))+t(G), where 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.
Related
- DL, DQ Spectrum and Eigenvalue-free Interval of Cayley Graphs over Finite Nilpotent Groups
- Chordality of Enhanced Power Graphs of Finite Simple Groups: A Resolution of a Conjecture
- On covering systems over finite nilpotent groups and lattices
- Finite bases for full power semirings of finite nilpotent semigroups
- Finite groups with simple or p-nilpotent maximal subgroups
BibTeX
@misc{chandran2026boxicity,
title = {{Boxicity of Power Graphs of Finite Nilpotent Groups}},
author = {L. Sunil Chandran and Jinia Ghosh and Suraj Kumar Sahoo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06816},
eprint = {2611.06816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06816},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boxicity of Power Graphs of Finite Nilpotent Groups
AU - L. Sunil Chandran
AU - Jinia Ghosh
AU - Suraj Kumar Sahoo
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06816
UR - https://arxiv.org/abs/2611.06816
AB - The \textit{boxicity} of an undirected graph Γ, denoted as 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 ℝ^d. The \emph{power graph} of a group G, denoted as 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=∏_i=1^rp_i^e_i>1, where the p_i are distinct primes, we prove that its power graph has boxicity ∑_i=1^re_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 box(Pow(G))=box(Pow(ℤ_exp(G)))+t(G), where 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.
ER -