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2009 Contributions to Seymour's second neighborhood conjecture
James Brantner, Greg Brockman, Bill Kay, Emma Snively
Involve 2(4): 387-395 (2009). DOI: 10.2140/involve.2009.2.387

Abstract

Let D be a simple digraph without loops or digons. For any vV(D) let N1(v) be the set of all nodes at out-distance 1 from v and let N2(v) be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist some vV(D) such that |N1(v)||N2(v)|. We provide several properties a “minimal” graph which does not contain such a node must have. Moreover, we show that if one such graph exists, then there exist infinitely many.

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James Brantner. Greg Brockman. Bill Kay. Emma Snively. "Contributions to Seymour's second neighborhood conjecture." Involve 2 (4) 387 - 395, 2009. https://doi.org/10.2140/involve.2009.2.387

Information

Received: 18 August 2008; Revised: 3 August 2009; Accepted: 13 August 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1194.05049
MathSciNet: MR2579558
Digital Object Identifier: 10.2140/involve.2009.2.387

Subjects:
Primary: 05C20‎

Rights: Copyright © 2009 Mathematical Sciences Publishers

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