Truncated degree AT-orientations of outerplanar graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912172520505344 |
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| author | Deng, Chenglong Zhu, Xuding |
| author_facet | Deng, Chenglong Zhu, Xuding |
| contents | An AT-orientation of a graph $G$ is an orientation $D$ of $G$ such that the number of even Eulerian sub-digraphs and the number of odd Eulerian sub-digraphs of $D$ are distinct. Given a mapping $f: V(G) \to \mathbb{N}$, we say $G$ is $f$-AT if $G$ has an AT-orientation $D$ with $ < f(v)$ for each vertex $v$. For a positive integer $k$, we say $G$ is $k$-truncated degree-AT if $G$ is $f$-AT for the mapping $f$ defined as $f(v) = \min #{k, d_G(v)#} $. This paper proves that 2-connected outerplanar graphs other than odd cycles are $5$-truncated degree-AT, and 2-connected bipartite outerplanar graphs are $4$-truncated degree-AT. As a consequence, 2-connected outerplanar graphs other than odd cycles are $5$-truncated degree paintable, and 2-connected bipartite outerplanar graphs are $4$-truncated degree paintable.
This improves the result of Hutchinson in [On list-coloring outerplanar graphs], where it was proved that maximal 2-connected outerplanar graphs other than are 5-truncated degree-choosable, and 2-connected bipartite outerplanar graphs are 4-truncated degree-choosable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Truncated degree AT-orientations of outerplanar graphs Deng, Chenglong Zhu, Xuding Combinatorics An AT-orientation of a graph $G$ is an orientation $D$ of $G$ such that the number of even Eulerian sub-digraphs and the number of odd Eulerian sub-digraphs of $D$ are distinct. Given a mapping $f: V(G) \to \mathbb{N}$, we say $G$ is $f$-AT if $G$ has an AT-orientation $D$ with $ < f(v)$ for each vertex $v$. For a positive integer $k$, we say $G$ is $k$-truncated degree-AT if $G$ is $f$-AT for the mapping $f$ defined as $f(v) = \min #{k, d_G(v)#} $. This paper proves that 2-connected outerplanar graphs other than odd cycles are $5$-truncated degree-AT, and 2-connected bipartite outerplanar graphs are $4$-truncated degree-AT. As a consequence, 2-connected outerplanar graphs other than odd cycles are $5$-truncated degree paintable, and 2-connected bipartite outerplanar graphs are $4$-truncated degree paintable. This improves the result of Hutchinson in [On list-coloring outerplanar graphs], where it was proved that maximal 2-connected outerplanar graphs other than are 5-truncated degree-choosable, and 2-connected bipartite outerplanar graphs are 4-truncated degree-choosable. |
| title | Truncated degree AT-orientations of outerplanar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.20811 |