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arXiv · 1711.01189

Vertices with the Second Neighborhood Property in Eulerian Digraphs

Michael Cary
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Abstract

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.

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Michael Cary. 2017-11-03. Vertices with the Second Neighborhood Property in Eulerian Digraphs. https://doi.org/10.7494/opmath.2019.39.6.765

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