A container theorem for general digraphs with forbidden subdigraphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909054737055744 |
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| author | Liang, Meili Guan, Yue Zheng, Ruiling Liu, Jianxi |
| author_facet | Liang, Meili Guan, Yue Zheng, Ruiling Liu, Jianxi |
| contents | In a seminal work, Kühn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for oriented graphs, does not directly extend to all digraphs due to the existence of counterexamples such as the double triangle $DK_3$. In this paper we prove a container theorem for general digraphs under a natural sparsity condition. For the edge-weight parameter $a=2$, this condition permits digraphs with $2$-cycles (density at most $1$) but excludes denser obstructions like $DK_3$; for larger $a$ it allows digraphs with a controlled density of $2$-cycles. As applications, we obtain asymptotic counting results for $H$-free digraphs and describe the typical structure of digraphs avoiding a fixed digraph $H$ satisfying our condition. Our results unify and extend several previous results in the area. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18542 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A container theorem for general digraphs with forbidden subdigraphs Liang, Meili Guan, Yue Zheng, Ruiling Liu, Jianxi Combinatorics 05C20, 05C35, 05C30, 05C65 In a seminal work, Kühn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for oriented graphs, does not directly extend to all digraphs due to the existence of counterexamples such as the double triangle $DK_3$. In this paper we prove a container theorem for general digraphs under a natural sparsity condition. For the edge-weight parameter $a=2$, this condition permits digraphs with $2$-cycles (density at most $1$) but excludes denser obstructions like $DK_3$; for larger $a$ it allows digraphs with a controlled density of $2$-cycles. As applications, we obtain asymptotic counting results for $H$-free digraphs and describe the typical structure of digraphs avoiding a fixed digraph $H$ satisfying our condition. Our results unify and extend several previous results in the area. |
| title | A container theorem for general digraphs with forbidden subdigraphs |
| topic | Combinatorics 05C20, 05C35, 05C30, 05C65 |
| url | https://arxiv.org/abs/2603.18542 |