Euler Tours in Hypergraphs

We show that a quasirandom k -uniform hypergraph G has a tight Euler tour subject to the necessary condition that k divides all vertex degrees. The case when G is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the k -subsets of an n -set.

[1]  Deryk Osthus,et al.  Hypergraph $F$-designs for arbitrary $F$ , 2017, 1706.01800.

[2]  Pawel Rzazewski,et al.  Tight Euler tours in uniform hypergraphs - computational aspects , 2017, Discret. Math. Theor. Comput. Sci..

[3]  J. Robert Johnson,et al.  Universal cycles for permutations , 2007, Discret. Math..

[4]  Yi Zhao,et al.  Recent advances on Dirac-type problems for hypergraphs , 2015, 1508.06170.

[5]  Zsolt Baranyai.  The edge-coloring of complete hypergraphs I , 1979, J. Comb. Theory, Ser. B.

[6]  Glenn H. Hurlbert,et al.  Near-Universal Cycles for Subsets Exist , 2009, SIAM J. Discret. Math..

[7]  Fan Chung Graham,et al.  Universal cycles for combinatorial structures , 1992, Discret. Math..

[8]  Vojtech Rödl,et al.  An approximate Dirac-type theorem for k-uniform hypergraphs , 2008, Comb..

[9]  Peter Keevash,et al.  The existence of designs II , 2018, 1802.05900.

[10]  Daniela Kühn,et al.  The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$. , 2020 .

[11]  Zbigniew Lonc,et al.  On Tours that Contain All Edges of a Hypergraph , 2010, Electron. J. Comb..

[12]  Alan M. Frieze,et al.  Packing tight Hamilton cycles in 3-uniform hypergraphs , 2011, SODA '11.

[13]  Zbigniew Lonc,et al.  Universal Cycle Packings and Coverings for k-Subsets of an n-Set , 2016, Graphs Comb..

[15]  Alan M. Frieze,et al.  Packing hamilton cycles in random and pseudo-random hypergraphs , 2012, Random Struct. Algorithms.

[16]  Daniela Kühn,et al.  Decompositions of complete uniform hypergraphs into Hamilton Berge cycles , 2014, J. Comb. Theory, Ser. A.

[17]  Brett Stevens,et al.  Hamiltonian decompositions of complete k-uniform hypergraphs , 2010, Discret. Math..

[18]  Glenn H. Hurlbert,et al.  1-Overlap cycles for Steiner triple systems , 2014, Des. Codes Cryptogr..

[19]  Miroslaw Truszczynski,et al.  Optimal F-Graphs for the Family of all K-Subsets of an N-Set , 1981, FODO.

[20]  Simon R. Blackburn.  The existence of k-radius sequences , 2012, J. Comb. Theory, Ser. A.

[21]  Andrew Wagner,et al.  Triple Systems are Eulerian , 2017 .

[22]  Glenn H. Hurlbert.  On Universal Cycles for k-Subsets of an n-Set , 1994, SIAM J. Discret. Math..

[23]  Benjamin Doerr,et al.  Analyzing Randomized Search Heuristics: Tools from Probability Theory , 2011, Theory of Randomized Search Heuristics.

[24]  Bradley W. Jackson.  Universal cycles of k-subsets and k-permutations , 1993, Discret. Math..