Tight paths in fully directed hypergraphs

TL;DR

This paper investigates the existence and length of tight paths in fully directed hypergraphs, establishing bounds on parameters that guarantee long paths and providing exact values for small cases.

Contribution

It introduces bounds on the minimum k for which hypergraph paths grow with n, and determines exact values for small uniformities, advancing understanding of directed hypergraph structures.

Findings

Bounds on k for infinite path length growth in hypergraphs

Exact values of f(n,r,k) for r ≤ 5

Growth rate estimates for specific hypergraph parameters

Abstract

It is well-known that every tournament has a spanning path. We consider hypergraph analogues. In an \emph{$r$-uniform fully directed hypergraph}, or \emph{$r$-digraph}, every edge is a list or $r$ distinct vertices. An $(r,k)$-tournament is an $r$-digraph $G$ such that for every $r$-set $S$ of vertices in $G$, exactly $k$ of the orderings of $S$ are edges in $G$. A \emph{directed tight path} is an $r$-digraph $G$ whose vertices can be ordered so that the intervals of size $r$ are the edges in $G$. Let $f(n,r,k)$ be the maximum $s$ such that every $n$-vertex $(r,k)$-tournament contains a tight path on $s$ vertices. Since every tournament has a spanning path, we have $f(n,2,1)=n$. In this paper, we show that the minimum $k$ such that $f(n,r,k)$ tends to infinity with $n$ is in the interval $\left[\left(1-\frac{1}{r}-O(\frac{\log r}{r^2\log\log r})\right)r!, ~\left(1-\frac{1}{r} - \frac{\varphi(r)-1}{r!}\right)r!\right]$ where $\varphi(r)$ is the Euler Totient Function, and we find the exact value when $r\le 5$. We also show that $\Omega(\sqrt{\log n/\log \log n}) \le f(n,3,3) \le O(\log n)$ and $f(n,3,4)\ge \Omega(n^{1/5})$.