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arXiv · 2405.12025

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

Zejun Huang·Chenxi Yang
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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 $ex_{ori}(n,\overrightarrow{S_{k,1}})$.

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Zejun Huang, Chenxi Yang. 2024-05-20. Extremal oriented graphs avoiding 1-subdivision of an in-star. https://arxiv.org/abs/2405.12025

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