A note on graphs of $k$-colourings
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911786131783680 |
|---|---|
| author | Hogan, Emma Scott, Alex Tamitegama, Youri Tan, Jane |
| author_facet | Hogan, Emma Scott, Alex Tamitegama, Youri Tan, Jane |
| contents | For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings, and has been extensively studied from both extremal and probabilistic perspectives. In this note, we show that for every graph $G$, there exists $k$ such that $G$ is uniquely determined by its $k$-colouring graph, confirming two conjectures of Asgarli, Krehbiel, Levinson and Russell. We further show that no finite family of generalised chromatic polynomials for $G$, which encode induced subgraph counts of its colouring graphs, uniquely determine $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on graphs of $k$-colourings Hogan, Emma Scott, Alex Tamitegama, Youri Tan, Jane Combinatorics 05C31 (Primary) 05C15 (Secondary) For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings, and has been extensively studied from both extremal and probabilistic perspectives. In this note, we show that for every graph $G$, there exists $k$ such that $G$ is uniquely determined by its $k$-colouring graph, confirming two conjectures of Asgarli, Krehbiel, Levinson and Russell. We further show that no finite family of generalised chromatic polynomials for $G$, which encode induced subgraph counts of its colouring graphs, uniquely determine $G$. |
| title | A note on graphs of $k$-colourings |
| topic | Combinatorics 05C31 (Primary) 05C15 (Secondary) |
| url | https://arxiv.org/abs/2402.04237 |