Exact minimum degree thresholds for perfect matchings in uniform hypergraphs

Abstract Given positive integers k and l where 4 divides k and k / 2 ⩽ l ⩽ k − 1 , we give a minimum l-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, as well as the hypergraph removal lemma and a structural result of Keevash and Sudakov relating to the Turan number of the expanded triangle.

[1]  Vojtech Rödl,et al.  A Dirac-Type Theorem for 3-Uniform Hypergraphs , 2006, Combinatorics, Probability and Computing.

[2]  Benny Sudakov,et al.  On A Hypergraph Turán Problem Of Frankl , 2005, Comb..

[3]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[4]  David E. Daykin,et al.  Degrees giving independent edges in a hypergraph , 1981, Bulletin of the Australian Mathematical Society.

[5]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[6]  Andrzej Rucinski,et al.  Perfect matchings (and Hamilton cycles) in hypergraphs with large degrees , 2011, Eur. J. Comb..

[7]  Imdadullah Khan,et al.  Perfect Matchings in 3-Uniform Hypergraphs with Large Vertex Degree , 2011, SIAM J. Discret. Math..

[8]  Daniela Kühn,et al.  Matchings in hypergraphs of large minimum degree , 2006 .

[9]  Vojtech Rödl,et al.  Perfect matchings in uniform hypergraphs with large minimum degree , 2006, Eur. J. Comb..

[10]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[11]  Hiêp Hàn,et al.  On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree , 2009, SIAM J. Discret. Math..

[12]  Oleg Pikhurko,et al.  Perfect Matchings and K43-Tilings in Hypergraphs of Large Codegree , 2008, Graphs Comb..

[13]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[14]  Vojtech Rödl,et al.  Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels , 2011, J. Comb. Theory, Ser. A.

[15]  W. T. Gowers,et al.  Hypergraph regularity and the multidimensional Szemerédi theorem , 2007, 0710.3032.

[16]  Klas Markström,et al.  Perfect matchings in 3-partite 3-uniform hypergraphs , 2011, J. Comb. Theory, Ser. A.

[17]  Daniela Kühn,et al.  Matchings in 3-uniform hypergraphs , 2010, J. Comb. Theory, Ser. B.

[18]  Vojtech Rödl,et al.  Perfect matchings in large uniform hypergraphs with large minimum collective degree , 2009, J. Comb. Theory, Ser. A.

[19]  Jozef Skokan,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006 .

[20]  Vojtech Rödl,et al.  Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve) , 2010 .

[21]  Peter Frankl.  Asymptotic solution of a turán-type problem , 1990, Graphs Comb..

[22]  Klas Markström,et al.  $F$-factors in hypergraphs via absorption , 2011 .

[23]  Peter Keevash,et al.  ON A HYPERGRAPH TUR ´ AN PROBLEM OF FRANKL , 2005 .

[24]  D. Kuhn,et al.  Surveys in Combinatorics 2009: Embedding large subgraphs into dense graphs , 2009, 0901.3541.

[25]  Ron Aharoni,et al.  Perfect matchings in r-partite r-graphs , 2009, Eur. J. Comb..

[26]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[27]  Peter Keevash,et al.  A Geometric Theory for Hypergraph Matching , 2011, 1108.1757.

[28]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[29]  Daniela Kühn,et al.  Fractional and integer matchings in uniform hypergraphs , 2013, Eur. J. Comb..

[30]  Andrzej Czygrinow,et al.  Tight Co-Degree Condition for Perfect Matchings in 4-Graphs , 2012, Electron. J. Comb..