Contributions to Seymour's Second Neighborhood Conjecture
Combinatorics
2008-08-19 v3
Abstract
Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there must exist some v in V(D) such that |N_1(v)| is less than or equal to |N_2(v)|, as well as examine properties of a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex.
Keywords
Cite
@article{arxiv.0808.0946,
title = {Contributions to Seymour's Second Neighborhood Conjecture},
author = {James N. Brantner and Greg Brockman and Bill Kay and Emma E. Snively},
journal= {arXiv preprint arXiv:0808.0946},
year = {2008}
}
Comments
9 pages, 4 figures