Abstract
Abstract Sullivan stated the conjectures: (1) every oriented graph has a vertex x such that d ++(x) ≥ d −(x) and (2) every oriented graph has a vertex x such that d ++(x) + d + (x) ≥ 2d −(x). In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament D, there are at least two vertices satisfying (1) and either there exist two vertices satisfying (2) or there exists a vertex v satisfying d ++(v) + d + (v) ≥ 2d −(v) + 2 if D has no vertex of in-degree zero.
Highlights
In this paper, we consider finite digraphs without loops and multiple arcs
Sullivan stated the conjectures: (1) every oriented graph has a vertex x such that d++(x) ≥ d−(x) and (2) every oriented graph has a vertex x such that d++(x) + d+(x) ≥ 2d−(x). We prove that these conjectures hold for local tournaments
If V1 and V2 are disjoint subsets of vertices of D such that there is no arc from V2 to V1 and a → b for each a ∈ V1 and each b ∈ V2, we say that V1 completely dominates V2 and denote it by V1 ⇒ V2
Summary
We consider finite digraphs without loops and multiple arcs. The main source for terminology and notation is ref. [1]. (Seymour’s Second Neighborhood Conjecture (SSNC)) For any oriented graph D, there exists a vertex v in D such that d++(v) ≥ d+(v). Fidler and Yuster [5] further developed the median order approach and proved that SSNC holds for oriented graphs D with minimum degree |V(D)| − 2|, tournaments minus a star and tournaments minus the arc set of a subtournament. Gutin and Li [9] proved SSNC for extended tournaments and quasi-transitive oriented graphs. Li and Sheng [12,13] proved Sullivan’s Conjectures for tournaments, extended tournaments, quasitransitive oriented graphs, and bipartite tournaments They obtained the following results: Theorem 1.5.
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