Reconfiguration of List Colourings
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908362928553984 |
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| author | Cambie, Stijn van Batenburg, Wouter Cames Cranston, Daniel W. Heuvel, Jan van den Kang, Ross J. |
| author_facet | Cambie, Stijn van Batenburg, Wouter Cames Cranston, Daniel W. Heuvel, Jan van den Kang, Ross J. |
| contents | Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|\ge \mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $Δ$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $\mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to $\{1,\ldots,Δ+1\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reconfiguration of List Colourings Cambie, Stijn van Batenburg, Wouter Cames Cranston, Daniel W. Heuvel, Jan van den Kang, Ross J. Combinatorics Discrete Mathematics Data Structures and Algorithms Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|\ge \mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $Δ$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $\mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to $\{1,\ldots,Δ+1\}$. |
| title | Reconfiguration of List Colourings |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2505.08020 |