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The Bounds of the Edge Number in Generalized Hypertrees

1,2,3, 1,2,3,*, 2,3,4, 1,2,3 and 5
1
School of Computer, Qinghai Normal University, Xining 810008, China
2
Key Laboratory of Tibetan Information Processing and Machine Translation in QH, Xining 810008, China
3
Key Laboratory of the Education Ministry for Tibetan Information Processing, Xining 810008, China
4
School of Computer Science, Shannxi Normal university, Xi’an 710062, China
5
School of mathematics and statistics, Qinghai Normal University, Xining 810008, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(1), 2; https://doi.org/10.3390/math7010002
Received: 11 November 2018 / Revised: 14 December 2018 / Accepted: 15 December 2018 / Published: 20 December 2018
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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PDF [2405 KB, uploaded 20 December 2018]
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Abstract

A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T E is a fixed value k   ( 2 k r ) . We focus on the case that G H T E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n r + 1 edges if r 3   and   n 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T E has exactly k   ( 2 k r 1 ) components. View Full-Text
Keywords: hypergraph; generalized hypertree; bound; component hypergraph; generalized hypertree; bound; component
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Zhang, K.; Zhao, H.; Ye, Z.; Zhu, Y.; Wei, L. The Bounds of the Edge Number in Generalized Hypertrees. Mathematics 2019, 7, 2.

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