Weak diameter choosability of graphs with an excluded minor
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909597180100608 |
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| author | Crouch, Joshua Liu, Chun-Hung |
| author_facet | Crouch, Joshua Liu, Chun-Hung |
| contents | Weak diameter coloring of graphs recently attracted attention partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph $H$, $H$-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring $H$-minor free graphs with bounded weak diameter. As a corollary, $H$-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring.
When $H$ is planar, we prove a much stronger result: for every 2-list-assignment $L$ of an $H$-minor free graph, every precoloring with bounded weak diameter can be extended to an $L$-coloring with bounded weak diameter. As a corollary, for any planar graph $H$ and $H$-minor free graph $G$, there are exponentially many list-colorings of $G$ with bounded weak diameter (and with bounded clustering if $G$ also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering.
We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree $Δ$ are $k$-choosable with bounded weak diameter only when $k=Ω(\logΔ/\log\logΔ)$. On the other hand, we show that odd $H$-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd $H$-minor free graphs with bounded maximum degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17795 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak diameter choosability of graphs with an excluded minor Crouch, Joshua Liu, Chun-Hung Combinatorics Weak diameter coloring of graphs recently attracted attention partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph $H$, $H$-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring $H$-minor free graphs with bounded weak diameter. As a corollary, $H$-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring. When $H$ is planar, we prove a much stronger result: for every 2-list-assignment $L$ of an $H$-minor free graph, every precoloring with bounded weak diameter can be extended to an $L$-coloring with bounded weak diameter. As a corollary, for any planar graph $H$ and $H$-minor free graph $G$, there are exponentially many list-colorings of $G$ with bounded weak diameter (and with bounded clustering if $G$ also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering. We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree $Δ$ are $k$-choosable with bounded weak diameter only when $k=Ω(\logΔ/\log\logΔ)$. On the other hand, we show that odd $H$-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd $H$-minor free graphs with bounded maximum degree. |
| title | Weak diameter choosability of graphs with an excluded minor |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.17795 |