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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
School of Computer, Qinghai Normal University, Xining 810008, China
Key Laboratory of Tibetan Information Processing and Machine Translation in QH, Xining 810008, China
Key Laboratory of the Education Ministry for Tibetan Information Processing, Xining 810008, China
School of Computer Science, Shannxi Normal university, Xi’an 710062, China
School of mathematics and statistics, Qinghai Normal University, Xining 810008, China
Author to whom correspondence should be addressed.
Mathematics 2019, 7(1), 2;
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)
PDF [2405 KB, uploaded 20 December 2018]


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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