A Remark on the Second Neighborhood Problem
Combinatorics
2016-02-09 v1
Abstract
Seymour's second neighborhood conjecture states that every simple digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). We define "good" digraphs and prove a statement that implies that every feed vertex of a tournament has the SNP. In the case of digraphs missing a matching, we exhibit a feed vertex with the SNP by refining a proof due to Fidler and Yuster and using good digraphs. Moreover, in some cases we exhibit two vertices with SNP.
Cite
@article{arxiv.1509.03282,
title = {A Remark on the Second Neighborhood Problem},
author = {Salman Ghazal},
journal= {arXiv preprint arXiv:1509.03282},
year = {2016}
}
Comments
arXiv admin note: substantial text overlap with arXiv:1106.5463