Extremal digraphs containing at most $t$ paths of length 2 with the same endpoints
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 $g(n,t)=\left\lceil(n+t)/{2}\right\rceil \left\lfloor(n-t)/{2}\right\rfloor+tn+1$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.16101
- arXiv:
- arXiv:2406.16101
- Bibcode:
- 2024arXiv240616101H
- Keywords:
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- Mathematics - Combinatorics