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Asymptotic Sharpness of Bounds on Hypertrees

100%
Lin Y., Kang L., Shan E.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
789-795
EN
The hypertree can be defined in many different ways. Katona and Szabó introduced a new, natural definition of hypertrees in uniform hypergraphs and investigated bounds on the number of edges of the hypertrees. They showed that a k-uniform hypertree on n vertices has at most [...] (nk−1) $\left( {\matrix{n \cr {k - 1} } } \right)$ edges and they conjectured that the upper bound is asymptotically sharp. Recently, Szabó verified that the conjecture holds by recursively constructing an infinite sequence of k-uniform hypertrees and making complicated analyses for it. In this note we give a short proof of the conjecture by directly constructing a sequence of k-uniform k-hypertrees.
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Bounds on the Number of Edges of Edge-Minimal, Edge-Maximal and L-Hypertrees

75%
Szabó P. G. N.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 2
259-278
EN
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 [...] 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 [...] .
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