Abstract
The second neighborhood conjecture of Seymour says that every antisymmetric digraph has a vertex whose second neighborhood is not smaller than the first one. The Caccetta–Häggkvist conjecture says that every digraph with n vertices and minimum out-degree r contains a cycle of length at most ⌈n/r⌉. We give a proof of the former conjecture for digraphs with out-degree r and connectivity r−1, and of the second one for digraphs with connectivity r−1 and r≥n/3. The main tool is the isoperimetric method of Hamidoune.
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