Abstract: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.
From: Jianxi Liu [view email]
[v1]
Thu, 19 Mar 2026 06:57:40 UTC (7 KB)
[v2]
Tue, 19 May 2026 03:49:08 UTC (8 KB)
[v3]
Mon, 15 Jun 2026 04:18:18 UTC (13 KB)
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