A Minimum Counterexample Proof of the Seymour Second Neighborhood Conjecture via the Graph Level Order

Abstract

We provide a constructive proof of the Seymour Second Neighborhood Conjecture (SSNC) by reframing the problem as a set-packing optimization problem. The universal family of oriented graphs $\mathcal{O}$ is classified by their minimum out-degree $δ$. This shifts the objective to maximizing the number of non-Seymour vertices. A minimum counterexample (MCE) is a maximal packing of vertices that fail the SSNC. To prove such a packing is unsustainable, we introduce the Graph Level Order (GLOVER). This BFS-based coordinate system partitions $\mathcal{O}$ into rooted neighborhoods $R_i$ from a minimum out-degree node. Set-theoretic multiple parents resolve the double-counting that has plagued Seymour diamonds. This coordinate system also categorizes transitive triangles into eight distinct types and proves that seven are inconsistent in an MCE environment. Distinguishing it from BFS, the MCE environment forces cycles in the first neighborhood of every parent. These cause neighborhoods to become quadratically dense as they both decrease in size and need more arcs. The proof concludes with a supply-demand collision. Arc capacity is consumed when $i > \fracδ{3}$. This makes the packing of non-Seymour vertices unsustainable, forcing the appearance of a Seymour vertex in every graph of $\mathcal{O}$. The algorithm to identify these vertices is $O(|V|+|E|)$. This confirms that it can operate on large oriented networks that are dense and detectable in polynomial time.