# The second out-neighbourhood for local tournaments

## TL;DR

This paper proves Sullivan's conjectures regarding out-neighbourhoods in local tournaments, showing that certain degree inequalities hold for vertices in these graphs, with specific conditions and multiple vertices satisfying the conjectures.

## Contribution

It establishes the validity of Sullivan's conjectures for local tournaments and identifies conditions under which multiple vertices satisfy these degree inequalities.

## Key findings

- Sullivan's conjecture (1) holds for local tournaments with no in-degree zero.
- Sullivan's conjecture (2) holds or a stronger inequality is satisfied in local tournaments.
- At least two vertices satisfy the conjecture (1) in local tournaments without in-degree zero.

## 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 $d^{++}(x)+d^{+}(x)\geq 2d^{-}(x)$. 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 $d^{++}(v)+d^{+}(v)\geq 2d^{-}(v)+2$ if $D$ has no vertex of in-degree zero.