Subgraphs with a positive minimum semidegree in digraphs with large outdegree

TL;DR

This paper proves that large outdegree directed graphs contain subgraphs with positive minimum semidegree, providing an asymptotically optimal bound when outdegree is small relative to the number of vertices.

Contribution

It establishes a new lower bound on the minimum semidegree of subgraphs in directed graphs with large outdegree, and proves its asymptotic optimality for sparse graphs.

Findings

Existence of subgraphs with positive minimum semidegree proportional to outdegree

Asymptotic optimality of the bound for sparse graphs

Quantitative relationship between outdegree and subgraph semidegree

Abstract

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$ then this bound is asymptotically best possible.