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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2501.16233 |
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| _version_ | 1866909467048673280 |
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| author | Chandran, L. Sunil Ghosh, Jinia |
| author_facet | Chandran, L. Sunil Ghosh, Jinia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boxicity and Cubicity of Divisor Graphs and Power Graphs Chandran, L. Sunil Ghosh, Jinia Combinatorics 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. |
| title | Boxicity and Cubicity of Divisor Graphs and Power Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.16233 |