Bounds on the Number of Edges in Hypertrees

TL;DR

This paper introduces a new concept of hypertrees in hypergraphs, establishing bounds on their number of edges, which generalizes tree properties from graphs to hypergraphs.

Contribution

It defines hypertrees in hypergraphs using chains and cycle-free conditions, and derives bounds on their edge counts, extending classical tree results to hypergraphs.

Findings

Hypertrees have at least n-(k-1) edges, with few exceptions.

Hypertrees have at most binomial(n, k-1) edges.

The upper bound is asymptotically sharp for 3-uniform hypertrees.

Abstract

Let $\mathcal{H}$ be a $k$-uniform hypergraph. A chain in $\mathcal{H}$ is a sequence of its vertices such that every $k$ consecutive vertices form an edge. In 1999 Katona and Kierstead suggested to use chains in hypergraphs as the generalisation of paths. Although a number of results have been published on hamiltonian chains in recent years, the generalization of trees with chains has still remained an open area. We generalize the concept of trees for uniform hypergraphs. We say that a $k$-uniform hypergraph $\mathcal{F}$ is a hypertree if every two vertices of $\mathcal{F}$ are connected by a chain, and an appropriate kind of cycle-free property holds. An edge-minimal hypertree is a hypertree whose edge set is minimal with respect to inclusion. After considering these definitions, we show that a $k$-uniform hypertree on $n$ vertices has at least $n-(k-1)$ edges up to a finite number of exceptions, and it has at most $\binom{n}{k-1}$ edges. The latter bound is asymptotically sharp in 3-uniform case.