Extremal digraphs containing at most $t$ paths of length 2 with the same endpoints

TL;DR

This paper determines the maximum size of directed graphs that avoid having $t+1$ paths of length 2 sharing the same start and end points, providing exact bounds for large graphs based on parity conditions.

Contribution

It establishes exact extremal sizes for $P_{t+1,2}$-free digraphs, extending extremal graph theory to specific path-structure constraints.

Findings

Exact maximum size formulas for large $n$

Differentiation based on parity of $loor{(n-t)/2}$

Bounds are tight for sufficiently large graphs

Abstract

Given a positive integer $t$, let $P_{t,2}$ be the digraph consisting of $t$ directed paths of length 2 with the same initial and terminal vertices. In this paper, we study the maximum size of $P_{t+1,2}$-free digraphs of order $n$, which is denoted by $ex(n, P_{t+1,2})$. For sufficiently large $n$, we prove that $ex(n, P_{t+1})=g(n,t)$ when $\lfloor(n-t)/{2} \rfloor$ is odd and $ex(n, P_{t+1,2})\in \{g(n,t)-1, g(n,t)\}$ when $\lfloor(n-t)/{2} \rfloor$ is even, where $g(n,t)=\left\lceil(n+t)/{2}\right\rceil \left\lfloor(n-t)/{2}\right\rfloor+tn+1$.