On the second neighborhood conjecture of Seymour for regular digraphs with almost optimal connectivity

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$n$ vertices and minimum out-degree r$r$ contains a cycle of length at most ⌈n/r⌉$\lceil n/r\rceil$. We give a proof of the former conjecture for digraphs with out-degree r$r$ and connectivity r−1$r-1$, and of the second one for digraphs with connectivity r−1$r-1$ and r≥n/3$r\ge n/3$. The main tool is the isoperimetric method of Hamidoune.