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

Abstract

In their paper, Bounds on the Number of Edges in Hypertrees, G.Y. Katona and P.G.N. Szabó 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 $\left(\genfrac{}{}{0ex}{}{n}{k-1}\right)$ 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 $\frac{1}{k-1}\left(\genfrac{}{}{0ex}{}{n}{2}\right)$.

arXiv comment19 pages, accepted by Discussiones Mathematicae Graph Theory in 2015; fixed proof of Theorem 6, added square at the end of proofs