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The number of connected sparsely edged uniform hypergraphs

Michakl Karoński, Tomasz kLuczak
1997 Discrete Mathematics  
Certain families of d-uniform hypergraphs are counted. In particular, the number of connected d-uniform hypergraphs with r vertices and r + k hyperedges, where k = o(log r/log tog r), is found.  ...  A hypergraph H is a pair (V,E) , where the set of hyperedges E is the family of subset of the vertex set V. A sequence of voelvl ...ekvk, where v; are vertices of H,  ...  [6] , the structure of a 'typical' connected graphs with n vertices and n + k edges has also been examined. Much less is known about the number of hypergraphs of a prescribed size.  ... 
doi:10.1016/s0012-365x(96)00076-3 fatcat:dydcuzj5dvgm3dr72exuq3uena

Sparse Hypergraphs and Pebble Game Algorithms [article]

Ileana Streinu, Louis Theran
2007 arXiv   pre-print
We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges.  ...  We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behaviour in terms of the sparsity parameters k and ℓ.  ...  The out-degree of a vertex is the number of edges which identify it as the tail and connect v to V − v; the in-degree is the number of edges that do not identify it as the tail.  ... 
arXiv:math/0703921v1 fatcat:iwctiihw6rhxxmhzta7ansb46e

Sparse hypergraphs and pebble game algorithms

Ileana Streinu, Louis Theran
2009 European journal of combinatorics (Print)  
We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges.  ...  We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and .  ...  The out-degree of a vertex is the number of edges which identify it as the tail and connect v to V − v; the in-degree is the number of edges that do not identify it as the tail.  ... 
doi:10.1016/j.ejc.2008.12.018 fatcat:h53pzoivcfdndoosb2rguplnnq

Toric algebra of hypergraphs [article]

Sonja Petrović, Despina Stasi
2013 arXiv   pre-print
The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph.  ...  We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs.  ...  Acknowledgements The authors thank György Turán for inspiring discussions, at the start of this project, about the fundamental problem of walks in hypergraphs; and also Elizabeth Gross for suggesting the  ... 
arXiv:1206.1904v3 fatcat:53eyqp4o7fapvmrpk5hdcixlga

Asymptotic growth of sparse saturated structures is locally determined

Zsolt Tuza
1992 Discrete Mathematics  
., Asymptotic growth of sparse saturated structures is locally determined, Discrete Mathematics 108 (1992) 397-402.  ...  The 'dual' of Twain's problem Let 9 and X be r-uniform hypergraphs without multiple edges, and denote by #(9 < X) the number of subhypergraphs isomorphic to 9 in X.  ...  We introduce some new hypergraph invariants, called local density and local sparseness of a hypergraph .F and prove that their values completely determine the order of magnitude of the smallest number  ... 
doi:10.1016/0012-365x(92)90692-9 fatcat:7jws22nhpbbcjl3hh4ouhfkfbq

Toric algebra of hypergraphs

Sonja Petrović, Despina Stasi
2013 Journal of Algebraic Combinatorics  
The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph.  ...  We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs.  ...  Acknowledgements The authors thank György Turán for inspiring discussions, at the start of this project, about the fundamental problem of walks in hypergraphs; and also Elizabeth Gross for suggesting the  ... 
doi:10.1007/s10801-013-0444-y fatcat:jd63bo7j55aelezckllmxpqn3e

Page 6541 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews  
An r-uniform hypergraph / is strongly ¥ -saturated if for every r-element set C not in # (that is, C is an edge of the comple- ment of # with respect to the complete r-uniform hypergraph) the number of  ...  .¥) and ssat(n,.F ) denote the minimum number of edges in a weakly 7 - saturated and a strongly ¥-saturated r-uniform hypergraph on n vertices, respectively.  ... 

A Sparse Model of Quantum Holography [article]

Shenglong Xu, Leonard Susskind, Yuan Su, Brian Swingle
2020 arXiv   pre-print
The resulting model has a new parameter, k, defined as the ratio of the number of terms in the Hamiltonian to the number of degrees of freedom, with the sparse limit corresponding to the thermodynamic  ...  We study a sparse version of the Sachdev-Ye-Kitaev (SYK) model defined on random hypergraphs constructed either by a random pruning procedure or by randomly sampling regular hypergraphs.  ...  In a d-regular, q-uniform hypergraph, the total number of terms is dN/q, so we set d = kq to keep the number of total terms equal to kN .  ... 
arXiv:2008.02303v1 fatcat:koh3svfzurhtxkf2aguuhh6zqi

Natural realizations of sparsity matroids

Ileana Streinu, Louis Theran
2011 Ars Mathematica Contemporanea  
structure of the underlying hypergraph G.  ...  Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k, )-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence  ...  A hypergraph G is defined to be d-uniform if all the edges have d endpoints.  ... 
doi:10.26493/1855-3974.197.461 fatcat:k5idpgif3jde7al6w22mnz6f7a

Natural realizations of sparsity matroids [article]

Ileana Streinu, Louis Theran
2010 arXiv   pre-print
structure of the underlying hypergraph G.  ...  Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence  ...  In Section 4, we describe two extensions of Theorem A: to non-uniform (k, ℓ)-sparse hypergraphs and to (k, ℓ)-graded-sparse hypergraphs.  ... 
arXiv:0711.3013v5 fatcat:ne3zthmvhzh4ngajxjvmjhe7m4

Probabilistic analysis of strong hypergraph coloring algorithms and the strong chromatic number

Jeanette P Schmidt
1987 Discrete Mathematics  
In these spaces the number of colors used by our algorithms is almost surely within a small constant factor (less than 4) of the strong chromatic number of the hypergraph. 0012-365X/87/$3.50 © 198"/, Elsevier  ...  We present coloring algorithms for several strong coloring problems and analyze their performance in spaces of random hypergraphs.  ...  Acknowledgment I would like to thank Eli Shamir for many discussions on the subject.  ... 
doi:10.1016/0012-365x(87)90101-4 fatcat:utsx5xodindb3bt4zq62sjofay

Higher order interactions destroy phase transitions in Deffuant opinion dynamics model [article]

Hendrik Schawe, Laura Hernández
2021 arXiv   pre-print
We show that including higher order interactions induces a drastic change in the onset of consensus for random hypergraphs; instead of the sharp phase transition, characteristic of the dyadic Deffuant  ...  The generalized model is naturally encoded in a hypergraph. We study this dynamics in different hypergraph topologies, from random hypergraph ensembles, to spatially embedded hyper-lattices.  ...  ACKNOWLEDGMENTS The authors acknowledge the OpLaDyn grant obtained in the 4th round of the Trans-Atlantic Platform Digging into Data Challenge (2016-147 ANR OPLADYN TAP-DD2016) and Labex MME-DII (Grant  ... 
arXiv:2111.12165v1 fatcat:pocaqut35vhejgsvu4o33xbbhu

Consistency of spectral hypergraph partitioning under planted partition model

Debarghya Ghoshdastidar, Ambedkar Dukkipati
2017 Annals of Statistics  
Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences.  ...  In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model.  ...  The model generates a sequence of planted sparse uniform hypergraphs, which taken together, represent the random hypergraph.  ... 
doi:10.1214/16-aos1453 fatcat:5lzpu4ygzvg63esublpwl3mipm

Coloring sparse hypergraphs [article]

Jeff Cooper, Dhruv Mubayi
2014 arXiv   pre-print
Fix k ≥ 3, and let G be a k-uniform hypergraph with maximum degree Δ. Suppose that for each l = 2, ..., k-1, every set of l vertices of G is in at most Δ^(k-l)/(k-1)/f edges.  ...  Then the chromatic number of G is O( (Δ/ f)^1/(k-1)). This extends results of Frieze and the second author and Bennett and Bohman.  ...  The degree of a vertex u ∈ V (G) is the number of edges containing that vertex. The maximum degree of a hypergraph G is the maximum degree of a vertex v ∈ V (G).  ... 
arXiv:1404.2895v1 fatcat:5ngshikawrewxjvlmlppojnrmu

Zero-One Law for random uniform hypergraphs [article]

Aleksandr Matushkin
2016 arXiv   pre-print
In this work limit probabilities of first-order properties of the random s-uniform hypergraph in the binomial model G^s(n,p) are studied.  ...  Moreover, for any rational ρ≥ 1/(s-1) we prove the existence of a strictly balanced s-uniform hypergraph with the density ρ.  ...  An s-uniform hypergraph, or s-hypergraph, G is a pair (V (G), E(G)) consisting of two sets, the vertices V (G) and edges E(G) of G, where each edge e ∈ E(G) is a set of s elements of V (G).  ... 
arXiv:1607.07654v1 fatcat:d56xwmitjzatrlmndmnbwxlc2i
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