Vertex Tur\'an problems for the oriented hypercube

TL;DR

This paper investigates the maximum size of vertex subsets in an oriented hypercube avoiding specific directed subgraphs, providing exact and asymptotic results for paths, cherries, and trees.

Contribution

It offers exact and asymptotic solutions for the oriented vertex Turán problem in hypercubes for various directed graphs, advancing combinatorial understanding.

Findings

Exact value for directed paths

Exact value for directed cherries

Asymptotic value for directed trees

Abstract

In this short note we consider the oriented vertex Tur\'an problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the directed cherry $\overrightarrow{V_2}$ and the asymptotic value of $ex_v(\overrightarrow{T}, \overrightarrow{Q_n})$ for any directed tree $\overrightarrow{T}$.