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

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$\acute{\mathrm{a}}$n number $e{x}_{ori}(n,F)$ is the maximum number of arcs in an $F$-free oriented graph of order $n$. In this paper, we investigate the oriented Tur$\acute{\mathrm{a}}$n number $e{x}_{ori}(n,\overrightarrow{S_{k,1}})$, where $\overrightarrow{S_{k,1}}$ is the $1$-subdivision of the in-star of order $k+1$. We determine $e{x}_{ori}(n,\overrightarrow{S_{k,1}})$ 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 $e{x}_{ori}(n,\overrightarrow{S_{k,1}})$.