Extending some results on the second neighborhood conjecture

TL;DR

This paper proves the Second Neighborhood Conjecture for specific classes of oriented graphs, extending previous results and showing that certain structural conditions guarantee the existence of vertices with large second neighborhoods.

Contribution

It generalizes known results by proving the conjecture for graphs with missing edges partitioned into matchings and stars, and for graphs partitioned into an independent set and a 2-degenerate graph.

Findings

Oriented graphs with missing edges as matchings and stars satisfy the conjecture.

Every such graph without a sink has at least two vertices with large second neighborhoods.

The conjecture holds for graphs partitioned into an independent set and a 2-degenerate graph.

Abstract

A vertex in a directed graph is said to have a large second neighborhood if it has at least as many second out-neighbors as out-neighbors. The Second Neighborhood Conjecture, first stated by Seymour, asserts that there is a vertex having a large second neighborhood in every oriented graph (a directed graph without loops or digons). We prove that oriented graphs whose missing edges can be partitioned into a (possibly empty) matching and a (possibly empty) star satisfy this conjecture. This generalizes a result of Fidler and Yuster. An implication of our result is that every oriented graph without a sink and whose missing edges form a (possibly empty) matching has at least two vertices with large second neighborhoods. This is a strengthening of a theorem of Havet and Thomasse, who showed that the same holds for tournaments without a sink. Moreover, we also show that the conjecture is true for oriented graphs whose vertex set can be partitioned into an independent set and a 2-degenerate graph.