Extremal oriented graphs avoiding 1-subdivision of an in-star

TL;DR

This paper studies the maximum number of arcs in large oriented graphs that avoid a specific subdivided in-star structure, providing exact results for small cases and bounds for larger ones.

Contribution

It determines the exact oriented Turán numbers for the 1-subdivision of in-stars of sizes 2 and 3, and establishes bounds for larger sizes.

Findings

Exact extremal numbers for k=2,3

Bounds for k≥4

Characterization of extremal graphs for small cases

Abstract

An oriented graph is a digraph obtained from an undirected graph by choosing an orientation for each edge. Given a positive integer $n$ and an oriented graph $F$, the oriented Tur$\acute{\rm a}$n number $ex_{ori}(n,F)$ is the maximum number of arcs in an $F$-free oriented graph of order $n$. In this paper, we investigate the oriented Tur$\acute{\rm a}$n number $ex_{ori}(n, \overrightarrow{S_{k,1}} )$, where $\overrightarrow{S_{k,1}}$ is the $1$-subdivision of the in-star of order $k+1$. We determine $ex_{ori}(n,\overrightarrow{S_{k,1}}) $ for $k=2,3$ as well as the extremal oriented graphs. For $k\ge 4$, we establish a lower bound and an upper bound on $ex_{ori}(n,\overrightarrow{S_{k,1}})$.