Vertices with the Second Neighborhood Property in Eulerian Digraphs

TL;DR

This paper proves that certain Eulerian digraphs with specific cycle intersection properties satisfy the Second Neighborhood Conjecture and can contain Seymour vertices, advancing understanding of neighborhood properties in directed graphs.

Contribution

It introduces a digraph variant of cycle intersection graphs and demonstrates their role in confirming the Second Neighborhood Conjecture for specific Eulerian digraphs.

Findings

Eulerian digraphs with simple cycle intersection graphs satisfy the Second Neighborhood Conjecture.

Local simplicity conditions can imply the existence of Seymour vertices.

The approach links cycle intersection structures to neighborhood properties in digraphs.

Abstract

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.