Truncated degree DP-colourability of $K_{2,4}$-minor free graphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929743370125312 |
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| author | Lo, On-Hei Solomon Wang, Cheng Zhou, Huan Zhu, Xuding |
| author_facet | Lo, On-Hei Solomon Wang, Cheng Zhou, Huan Zhu, Xuding |
| contents | Assume $G$ is a graph and $k$ is a positive integer. Let $f$ from $V(G)$ to $ N$ be defined as $f(v)$ is the minimum of $k$ and $d(v)$. If $G$ is $f$-DP-colourable (respectively, $f$-choosable), then we say $G$ is $k$-truncated degree DP-colourable (respectively, $k$-truncated degree-choosable). Hutchinson proved that 2-connected maximal outerplanar graphs other than the triangle are $5$-truncated degree-choosable, and asked whether the result can be extended to all outerplanar graphs, and the question remained open.
This paper proves that 2-connected $K24$-minor free graphs other than cycles and complete graphs are $5$-truncated degree DP-colourable. This not only answers Hutchinson's question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15962 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Truncated degree DP-colourability of $K_{2,4}$-minor free graphs Lo, On-Hei Solomon Wang, Cheng Zhou, Huan Zhu, Xuding Combinatorics Assume $G$ is a graph and $k$ is a positive integer. Let $f$ from $V(G)$ to $ N$ be defined as $f(v)$ is the minimum of $k$ and $d(v)$. If $G$ is $f$-DP-colourable (respectively, $f$-choosable), then we say $G$ is $k$-truncated degree DP-colourable (respectively, $k$-truncated degree-choosable). Hutchinson proved that 2-connected maximal outerplanar graphs other than the triangle are $5$-truncated degree-choosable, and asked whether the result can be extended to all outerplanar graphs, and the question remained open. This paper proves that 2-connected $K24$-minor free graphs other than cycles and complete graphs are $5$-truncated degree DP-colourable. This not only answers Hutchinson's question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability. |
| title | Truncated degree DP-colourability of $K_{2,4}$-minor free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.15962 |