Opuscula Math. 39, no. 6 (2019), 765-772
Vertices with the second neighborhood property in Eulerian digraphs
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.
Mathematics Subject Classification: 05C45, 05C12, 05C20.
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- Communicated by Andrzej Żak.
- Received: 2019-04-19.
- Revised: 2019-10-02.
- Accepted: 2019-10-04.
- Published online: 2019-11-22.