On offset Hamilton cycles in random hypergraphs

Abstract Let k and l be integers satisfying 1 ≤ l ≤ k ∕ 2 . An l -offset Hamilton cycle C in a k -uniform hypergraph H on  n vertices is a collection of edges of H such that for some cyclic order of [ n ] and every even i every pair of consecutive edges E i − 1 , E i in C (in the natural ordering of the edges) satisfies | E i − 1 ∩ E i | = l and every pair of consecutive edges E i , E i + 1 in C satisfies | E i ∩ E i + 1 | = k − l . We show that in general e k l ! ( k − l ) ! ∕ n k is the sharp threshold for the existence of the l -offset Hamilton cycle in the random k -uniform hypergraph H n , p ( k ) . We also examine this structure’s natural connection to the 1–2–3 Conjecture.

[1]  J. Komlos,et al.  First Occurrence of Hamilton Cycles in Random Graphs , 1985 .

[2]  J. Kahn,et al.  Factors in random graphs , 2008 .

[3]  L. Pósa,et al.  Hamiltonian circuits in random graphs , 1976, Discret. Math..

[4]  János Komlós,et al.  Limit distribution for the existence of Hamiltonian cycles in a random graph , 2006, Discret. Math..

[5]  Ben Seamone,et al.  The 1-2-3 Conjecture and related problems: a survey , 2012, ArXiv.

[6]  Bruce A. Reed,et al.  Degree constrained subgraphs , 2005, Discret. Appl. Math..

[7]  Yury Person,et al.  Spanning structures and universality in sparse hypergraphs , 2015, Random Struct. Algorithms.

[8]  Asaf Ferber,et al.  Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs , 2015, Electron. J. Comb..

[9]  A. Thomason,et al.  Edge weights and vertex colours , 2004 .

[10]  Andrzej Dudek,et al.  Optimal Divisibility Conditions for Loose Hamilton Cycles in Random Hypergraphs , 2012, Electron. J. Comb..

[11]  Alan M. Frieze,et al.  Loose Hamilton Cycles in Random 3-Uniform Hypergraphs , 2010, Electron. J. Comb..

[12]  A. Frieze,et al.  Introduction to Random Graphs , 2016 .

[13]  Daniel Poole,et al.  On Weak Hamiltonicity of a Random Hypergraph , 2014, 1410.7446.

[14]  Florian Pfender,et al.  Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture , 2010, J. Comb. Theory B.

[15]  Andrzej Dudek,et al.  Tight Hamilton cycles in random uniform hypergraphs , 2011, Random Struct. Algorithms.

[16]  Bruce A. Reed,et al.  Vertex-Colouring Edge-Weightings , 2007, Comb..

[17]  Qinglin Yu,et al.  On vertex-coloring 13-edge-weighting , 2008 .

[18]  Colin McDiarmid.  General percolation and random graphs , 1981, Advances in Applied Probability.

[19]  Florian Pfender,et al.  The 1‐2‐3‐Conjecture for Hypergraphs , 2013, J. Graph Theory.

[20]  Andrzej Dudek,et al.  Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs , 2015, Electron. J. Comb..