On the maximum induced density of directed stars and related problems

TL;DR

This paper proves that the maximum induced density of directed stars in a directed graph is achieved by an iterated blow-up construction, confirming a conjecture and providing insights into extremal density problems.

Contribution

It establishes the extremality of an iterated blow-up construction for directed stars, extending previous results and addressing a key conjecture in graph density theory.

Findings

Maximum induced density of directed stars achieved by iterated blow-up

Confirmed conjecture for all k >= 3

Explored inducibility of bipartite digraphs

Abstract

Let k>=3 be an integer, we prove that the maximum induced density of the k-vertex directed star in a directed graph is attained by an iterated blow-up construction. This confirms a conjecture by Falgas-Ravry and Vaughan, who proved this for k=3, 4. This question provides the first known instance of density problem for which one can prove extremality of an iterated blow-up construction. We also study the inducibility of complete bipartite digraphs and discuss other related problems.