Arboricity: An acyclic hypergraph decomposition problem motivated by database theory

The arboricity of a hypergraph H is the minimum number of acyclic hypergraphs that partition H. The determination of the arboricity of hypergraphs is a problem motivated by database theory. The exact arboricity of the complete k-uniform hypergraph of order n is previously known only for k@?{1,2,n-2,n-1,n}. The arboricity of the complete k-uniform hypergraph of order n is determined asymptotically when k=n-O(log^1^-^@dn), @d positive, and determined exactly when k=n-3. This proves a conjecture of Wang (2008) [20] in the asymptotic sense.

[1]  Haim Hanani,et al.  On Quadruple Systems , 1960, Canadian Journal of Mathematics.

[2]  Edgar M. Palmer,et al.  On the spanning tree packing number of a graph: a survey , 2001, Discret. Math..

[3]  Catriel Beeri,et al.  On the Desirability of Acyclic Database Schemes , 1983, JACM.

[4]  Mihalis Yannakakis,et al.  Algorithms for Acyclic Database Schemes , 1981, VLDB.

[5]  Catriel Beeri,et al.  Properties of acyclic database schemes , 1981, STOC '81.

[6]  Ronald Fagin,et al.  A simplied universal relation assumption and its properties , 1982, TODS.

[7]  V. Vu.  New bounds on nearly perfect matchings in hypergraphs: higher codegrees do help , 2000 .

[8]  Mike J. Grannell,et al.  Steiner systems S(5, 6, v) with v = 72 and 84 , 1998, Math. Comput..

[9]  H. Hanani,et al.  On steiner systems , 1964 .

[10]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[11]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[12]  Li,et al.  ON ACYCLIC AND CYCLIC HYPERGRAPHS , 2002 .

[13]  Ronald Fagin,et al.  Degrees of acyclicity for hypergraphs and relational database schemes , 1983, JACM.

[14]  Nathan Goodman,et al.  Syntactic Characterization of Tree Database Schemas , 1983, JACM.

[15]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[16]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[17]  Ian Anderson,et al.  TRIPLE SYSTEMS (Oxford Mathematical Monographs) , 2000 .

[18]  Jean-Claude Bermond,et al.  The α-Arboricity of Complete Uniform Hypergraphs , 2011, SIAM J. Discret. Math..

[19]  Jian-fang Wang,et al.  Enumeration of Maximum Acyclic Hypergraphs , 2002 .