Extending some results on the second neighborhood conjecture

Abstract

If in a directed graph, v is an out-neighbor of u and w is an out-neighbor of v but not of u, then w is said to be a second out-neighbor of u. 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). It is straightforward to see that the conjecture is true for any oriented graph whose underlying undirected graph is bipartite. We extend this to show that the conjecture holds for oriented graphs whose vertex set can be partitioned into an independent set and a 2-degenerate graph. Fisher proved the conjecture for tournaments and later Havet and Thomassé provided a different proof for the same using median orders of tournaments. Havet and Thomassé in fact showed the stronger statement that if a tournament contains no sink, then it contains at least two vertices with large second neighborhoods. Using their techniques, Fidler and Yuster showed that the conjecture remains true for tournaments from which either a matching or a star has been removed. We extend this result to show that the conjecture holds even for tournaments from which both a matching and a star have been removed. This implies that a tournament from which a matching has been removed contains either a sink or two vertices with large second neighborhoods.