10-list Recoloring of Planar Graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918159802433536 |
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| author | Cranston, Daniel W. |
| author_facet | Cranston, Daniel W. |
| contents | Fix a planar graph $G$ and a list-assignment $L$ with $|L(v)|=10$ for all $v\in V(G)$. Let $α$ and $β$ be $L$-colorings of $G$. A recoloring sequence from $α$ to $β$ is a sequence of $L$-colorings, beginning with $α$ and ending with $β$, such that each successive pair in the sequence differs in the color on a single vertex of $G$. We show that there exists a constant $C$ such that for all choices of $α$ and $β$ there exists a recoloring sequence $σ$ from $α$ to $β$ that recolors each vertex at most $C$ times. In particular, $σ$ has length at most $C|V(G)|$. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 10-list Recoloring of Planar Graphs Cranston, Daniel W. Combinatorics Fix a planar graph $G$ and a list-assignment $L$ with $|L(v)|=10$ for all $v\in V(G)$. Let $α$ and $β$ be $L$-colorings of $G$. A recoloring sequence from $α$ to $β$ is a sequence of $L$-colorings, beginning with $α$ and ending with $β$, such that each successive pair in the sequence differs in the color on a single vertex of $G$. We show that there exists a constant $C$ such that for all choices of $α$ and $β$ there exists a recoloring sequence $σ$ from $α$ to $β$ that recolors each vertex at most $C$ times. In particular, $σ$ has length at most $C|V(G)|$. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area. |
| title | 10-list Recoloring of Planar Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.00679 |