Bounds on the Number of Edges of Edge-minimal, Edge-maximal and $l$-hypertrees

TL;DR

This paper investigates bounds on the number of edges in various types of hypertrees within hypergraphs, providing asymptotic sharpness results, improvements on bounds, and new constructions for $k$-uniform hypertrees.

Contribution

It verifies the asymptotic sharpness of known bounds, improves upper bounds for 2-hypertrees, and introduces a general extension construction for $k$-uniform hypertrees.

Findings

Asymptotic sharpness of the $inom{n}{k-1}$ upper bound confirmed.

Improved upper bounds for 2-hypertrees established.

New extension construction for $k$-uniform hypertrees proposed.

Abstract

In their paper, Bounds on the Number of Edges in Hypertrees, G.Y. Katona and P.G.N. Szab\'o introduced a new, natural definition of hypertrees in $k$-uniform hypergraphs and gave lower and upper bounds on the number of edges. They also defined edge-minimal, edge-maximal and $l$-hypertrees and proved an upper bound on the edge number of $l$-hypertrees. In the present paper, we verify the asymptotic sharpness of the $\binom{n}{k-1}$ upper bound on the number of edges of $k$-uniform hypertrees given in the above mentioned paper. We also make an improvement on the upper bound of the edge number of $2$-hypertrees and give a general extension construction with its consequences. We give lower and upper bounds on the maximal number of edges of $k$-uniform edge-minimal hypertrees and a lower bound on the number of edges of $k$-uniform edge-maximal hypertrees. In the former case, the sharp upper bound is conjectured to be asymptotically $\frac{1}{k-1}\binom{n}{2}$.