On orientations with forbidden out-degrees
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911910120652800 |
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| author | Henderschedt, Owen McDonald, Jessica |
| author_facet | Henderschedt, Owen McDonald, Jessica |
| contents | Let $G$ be a $d$-regular graph and let $F\subseteq\{0, 1, 2, \ldots, d\}$ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F|<\tfrac{1}{2}d$, then $G$ should admit an $F$-avoiding orientation, i.e., an orientation where no out-degrees are in the forbidden list $F$. The conjecture is known for $d\leq 4$ due to work of Ma and Lu, and here we extend this to $d\leq 6$. The conjecture has also been studied in a generalized version, where $d, F$ are changed from constant values to functions $d(v), F(v)$ that vary over all $v\in V(G)$. We provide support for this generalized version by verifying it for some new cases, including when $G$ is 2-degenerate and when every $F(v)$ has some specific structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On orientations with forbidden out-degrees Henderschedt, Owen McDonald, Jessica Combinatorics Let $G$ be a $d$-regular graph and let $F\subseteq\{0, 1, 2, \ldots, d\}$ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F|<\tfrac{1}{2}d$, then $G$ should admit an $F$-avoiding orientation, i.e., an orientation where no out-degrees are in the forbidden list $F$. The conjecture is known for $d\leq 4$ due to work of Ma and Lu, and here we extend this to $d\leq 6$. The conjecture has also been studied in a generalized version, where $d, F$ are changed from constant values to functions $d(v), F(v)$ that vary over all $v\in V(G)$. We provide support for this generalized version by verifying it for some new cases, including when $G$ is 2-degenerate and when every $F(v)$ has some specific structure. |
| title | On orientations with forbidden out-degrees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.05095 |