Sullivan’s second neighbourhood conjecture

Prove that every oriented graph contains a vertex v for which the size of the second out-neighbourhood of v is at least the size of the in-neighbourhood of v; that is, show |N^+_2(v)| ≥ |N^-_1(v)|.

Background

Sullivan proposed an analogue of Seymour’s conjecture comparing the second out-neighbourhood to the in-neighbourhood. This variant coincides with Seymour’s conjecture on Eulerian orientations and is known to hold in some special classes (e.g., tournaments), but remains open in general. The paper develops the notion of Sullivan-tight orientations (equality case) and shows that (generalized) lexicographic products preserve this property.

References

In her survey on the Caccetta-H\"aggkvist conjecture, Sullivan proposed the following variation of Seymour’s second neighbourhood conjecture. Every oriented graph contains at least one vertex such that $|N+_2(v)| \geq |N{-}_1(v)|$.

— Seymour-tight orientations  (2603.29626 - Guo et al., 31 Mar 2026) in Section 7 (Sullivan’s conjecture)