The Turan problems of directed paths and cycles in digraphs

TL;DR

This paper determines the maximum number of edges in large directed graphs that avoid certain paths and cycles, providing exact extremal structures for all sizes and orientations, extending previous results on specific cycles.

Contribution

It precisely characterizes the maximum size and extremal digraphs avoiding directed paths and cycles, including all orientations of C4, for large and all graph sizes.

Findings

Exact maximum sizes for P_k-free digraphs for large n

Exact maximum sizes for C_k-free digraphs for all n

Characterization of extremal digraphs avoiding orientations of C4

Abstract

Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size for large $n$. For all $n$, we also determine the precise maximum size of $\overrightarrow{C_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size. In addition, Huang and Lyu [\textit{Discrete Math. 343(5) 2020}] characterized the extremal digraphs avoiding an orientation of $C_4$. For all other orientations of $C_4$, we also study the maximum size and the extremal digraphs avoiding them.