Some Extensions of Thomassen's Theorem to Longer Paths
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866929282735931392 |
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| author | Nevin, Joshua |
| author_facet | Nevin, Joshua |
| contents | Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. In this paper, we prove some results about partial $L$-colorings $ϕ$ of $C$ with the property that any extension of $ϕ$ to an $L$-coloring of $\textrm{dom}(ϕ)\cup V(P)$ extends to $L$-color all of $G$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12128 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some Extensions of Thomassen's Theorem to Longer Paths Nevin, Joshua Combinatorics 05C15 G.2.2 Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. In this paper, we prove some results about partial $L$-colorings $ϕ$ of $C$ with the property that any extension of $ϕ$ to an $L$-coloring of $\textrm{dom}(ϕ)\cup V(P)$ extends to $L$-color all of $G$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces. |
| title | Some Extensions of Thomassen's Theorem to Longer Paths |
| topic | Combinatorics 05C15 G.2.2 |
| url | https://arxiv.org/abs/2207.12128 |