@Article{Cary2019,
author = "Cary, Michael",
title = "Vertices with the second neighborhood property in Eulerian digraphs",
journal = "Opuscula Math.",
year = "2019",
volume = "39",
number = "6",
pages = "765--772",
doi = "https://doi.org/10.7494/OpMath.2019.39.6.765",
url = "https://doi.org/10.7494/OpMath.2019.39.6.765",
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 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.",
keywords = "Eulerian digraph, second neighborhood conjecture, cycle decomposition, cycle intersection graph",
issn = "1232-9274"
}