Towards a proof of Seymour’s Second Neighborhood Conjecture James N. Brantner, Greg Brockman, Bill Kay, Emma E. Snively November 9, 2021 Abstract Let D be a simple digraph without loops or digons. For any v ∈ V (D) let N 1 (v) be the set of all nodes at out-distance 1 from v and let N 2 (v) be the set of all nodes at out-distance 2. We provide conditions under which there must exist some v ∈ V (D) such that |N 1 (v)|≤|N 2 (v)|, as well as examine extremal properties in a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many. 1 Introduction For the purposes of this article, we consider only simple nonempty digraphs (those containing no loops or multiple edges and having a nonempty vertex set), unless stated otherwise. We also require that our digraphs contain no digons, that is, if D is a digraph then (u, v) ∈ E(D) ⇒ (v,u) / ∈ E(D). If i is a positive integer, we denote the i th neighborhood of a vertex u in D by N i,D (u)= {v ∈ V (D)|dist D (u, v)= i}, where dist D (u, v) is the length of the shortest directed path from u to v in D (if there is no directed path from u to v, we set dist D (u, v)= ∞). If D is clear from context, we simply write N i (u) and dist(u, v). We also may wish to consider the i th in-neighborhood of a node N −i (u)= {v ∈ V (D)|dist(v,u)= i}. In addition, if V ′ ⊆ V (D), we let D[V ′ ] be the subgraph of D induced by V ′ . Graph theorists will be familiar with the following conjecture by Seymour: Conjecture 1.1 (Seymour’s Second Neighborhood Conjecture). Let D be a directed graph. Then there exists a vertex v 0 ∈ V (D) such that |N 1 (v 0 )|≤|N 2 (v 0 )|. In 1995, Dean [2] conjectured this to be true when D is a tournament. Dean’s Conjecture was subsequently proven by Fisher [4] in 1996. Further, in their 2001 paper Kaneko and Locke [5] showed Conjecture 1.1 to be true if the minimum outdegree of vertices in D is less than 7, and Cohn, Wright, and Godbole [1] showed that it holds for random graphs almost always. And finally, in 2007 Fidler and Yuster [3] proved that Conjecture 1.1 holds for graphs with minimum out-degree |V (D)|− 2, tournaments minus a star, and tournaments minus a sub-tournament. While over the years there have been several attempts at a proof of Conjecture 1.1, none of these have yet been successful. For completeness, we introduce the related Caccetta-H¨aggkvist conjecture: 1 arXiv:0808.0946v1 [math.CO] 7 Aug 2008