Seymour's Second Neighborhood Conjecture

Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for dense incomplete graphs a series of results restricting the structure of the missing edges. This work proves the conjecture for every oriented graph of order $n = 2\delta + 2$, where $\delta$ is the minimum outdegree, with no prescribed structure on the missing edges; with Fisher's tournament theorem this gives every oriented graph satisfying $n \le 2\delta + 2$.

A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on the missing edges.