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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.21287 |
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| _version_ | 1866910902339502080 |
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| author | Sheats, Hannah |
| author_facet | Sheats, Hannah |
| contents | In 1974, Erdős asked the following question: given a graph $G$ and a directed graph $\vec{H}$, how many ways are there to orient the edges of $G$ such that it does not contain $\vec{H}$ as a subgraph? We denote this value by $D(G, \vec{H})$. Further, we let $D(n, \vec{H})$ denote the maximum of $D(G, \vec{H})$ over all graphs $G$ on $n$ vertices. In 2006, Alon and Yuster gave an exact answer when $\vec{H}$ is a tournament. In 2023, Bucić, Janzer, and Sudakov gave asymptotic answers for all directed graphs $\vec{H}$, and in the same paper, they gave an exact answer when $\vec{H}$ is a directed cycle. In this paper, we give a better bound for some specific non-bipartite directed graphs. Further, we obtain exact values of $D(G, \vec{H})$ for some small non-edge-critical directed graphs $\vec{H}$. Finally, for these graphs, we classify all graphs $G$ that attain the bound $D(G, \vec{H}) = D(n, \vec{H})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21287 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orientations of graphs omitting non-edge-critical directed graphs Sheats, Hannah Combinatorics In 1974, Erdős asked the following question: given a graph $G$ and a directed graph $\vec{H}$, how many ways are there to orient the edges of $G$ such that it does not contain $\vec{H}$ as a subgraph? We denote this value by $D(G, \vec{H})$. Further, we let $D(n, \vec{H})$ denote the maximum of $D(G, \vec{H})$ over all graphs $G$ on $n$ vertices. In 2006, Alon and Yuster gave an exact answer when $\vec{H}$ is a tournament. In 2023, Bucić, Janzer, and Sudakov gave asymptotic answers for all directed graphs $\vec{H}$, and in the same paper, they gave an exact answer when $\vec{H}$ is a directed cycle. In this paper, we give a better bound for some specific non-bipartite directed graphs. Further, we obtain exact values of $D(G, \vec{H})$ for some small non-edge-critical directed graphs $\vec{H}$. Finally, for these graphs, we classify all graphs $G$ that attain the bound $D(G, \vec{H}) = D(n, \vec{H})$. |
| title | Orientations of graphs omitting non-edge-critical directed graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.21287 |