Subdigraphs of prescribed size and outdegree

TL;DR

This paper disproves a 2006 conjecture by constructing large tournaments where every half-sized subdigraph has significantly lower minimum out-degree than expected, showing the problem's negative answer.

Contribution

It provides a counterexample to a longstanding open problem about subdigraphs with prescribed outdegree in directed graphs.

Findings

Existence of large tournaments with low outdegree in all half-sized subdigraphs

Counterexample disproving the conjecture from 2006

Quantitative bounds on outdegree in subdigraphs

Abstract

In 2006, Noga Alon raised the following open problem: Does there exist an absolute constant $c>0$ such that every $2n$-vertex digraph with minimum out-degree at least $s$ contains an $n$-vertex subdigraph with minimum out-degree at least $\frac{s}{2}-c$ ? In this note, we answer this natural question in the negative, by showing that for arbitrarily large values of $n$ there exists a $2n$-vertex tournament with minimum out-degree $s=n-1$, in which every $n$-vertex subdigraph contains a vertex of out-degree at most $\frac{s}{2}-\left(\frac{1}{2}+o(1)\right)\log_3(s)$.