Subdigraphs of prescribed size and out‐degree

Abstract

AbstractIn 2006, Noga Alon raised the following open problem: Does there exist an absolute constant such that every ‐vertex digraph with minimum out‐degree at least contains an ‐vertex subdigraph with minimum out‐degree at least ? In this note, we answer this natural question in the negative, by showing that for arbitrarily large values of there exists a ‐vertex tournament with minimum out‐degree , in which every ‐vertex subdigraph contains a vertex of out‐degree at most .