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arXiv · 2603.18542

A container theorem for general digraphs with forbidden subdigraphs

Meili Liang·Yue Guan·Ruiling Zheng·Jianxi Liu
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Abstract

In a seminal work, K\"uhn, 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.

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Meili Liang, Yue Guan, Ruiling Zheng, Jianxi Liu. 2026-03-19. A container theorem for general digraphs with forbidden subdigraphs. https://arxiv.org/abs/2603.18542

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