The second out-neighbourhood for local tournaments

DOI10.1515/MATH-2020-0026arXiv1812.01800MaRDI QIDQ6310728

Juanjuan Liang, Ruijuan Li

Publication date: 4 December 2018

Abstract: Sullivan stated the conjectures: (1) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)geq{d}^{-}(x)$; (2) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)+d^{+}(x)geq2d^{-}(x)$. In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament $D$, we prove that $D$ has at least two vertices satisfying $(1)$ if $D$ has no vertex of in-degree zero. And, for a local tournament $D$, we prove that either there exist two vertices satisfying $(2)$ or there exists a vertex $v$ satisfying $d^{++}(v)+d^{+}(v)geq2d^{-}(v)+2$ if $D$ has no vertex of in-degree zero.

Mathematics Subject Classification ID

Distance in graphs (05C12) Directed graphs (digraphs), tournaments (05C20)

This page was built for publication: The second out-neighbourhood for local tournaments

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6310728)