Hamiltonicity of Bell and Stirling Colour Graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911321962840064 |
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| author | Finbow, Stephen MacGillivray, Gary |
| author_facet | Finbow, Stephen MacGillivray, Gary |
| contents | For a graph $G$ and a positive integer $k$, the $k$-Bell colour graph of $G$ is the graph whose vertices are the partitions of $V$ into at most $k$ independent sets, with two of these being adjacent if there exists a vertex $x$ such that the partitions are identical when restricted to $V - \{x\}$. The $k$-Stirling Colour graph of $G$ is defined similarly, but for partitions into exactly $k$ independent sets.
We show that every graph on $n$ vertices, except $K_n$ and $K_n - e$, has a Hamiltonian $n$-Bell colour graph, and this result is best possible. It is also shown that, for $k \geq 4$, the $k$-Stirling colour graph of a tree with at least $k+1$ vertices is Hamiltonian, and the 3-Bell colour graph of a tree with at least 3 vertices is Hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hamiltonicity of Bell and Stirling Colour Graphs Finbow, Stephen MacGillivray, Gary Combinatorics 05C15 For a graph $G$ and a positive integer $k$, the $k$-Bell colour graph of $G$ is the graph whose vertices are the partitions of $V$ into at most $k$ independent sets, with two of these being adjacent if there exists a vertex $x$ such that the partitions are identical when restricted to $V - \{x\}$. The $k$-Stirling Colour graph of $G$ is defined similarly, but for partitions into exactly $k$ independent sets. We show that every graph on $n$ vertices, except $K_n$ and $K_n - e$, has a Hamiltonian $n$-Bell colour graph, and this result is best possible. It is also shown that, for $k \geq 4$, the $k$-Stirling colour graph of a tree with at least $k+1$ vertices is Hamiltonian, and the 3-Bell colour graph of a tree with at least 3 vertices is Hamiltonian. |
| title | Hamiltonicity of Bell and Stirling Colour Graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2512.13864 |