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Loose Hamilton Cycles in Regular Hypergraphs

Published online by Cambridge University Press:  24 September 2014

ANDRZEJ DUDEK
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA (e-mail: andrzej.dudek@wmich.edu)
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: alan@random.math.cmu.edu)
ANDRZEJ RUCIŃSKI
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: rucinski@amu.edu.pl)
MATAS ŠILEIKIS
Affiliation:
Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden (e-mail: matas.sileikis@math.uu.se)

Abstract

We establish a relation between two uniform models of random k-graphs (for constant k ⩾ 3) on n labelled vertices: ℍ (k) (n,m), the random k-graph with exactly m edges, and ℍ (k) (n,d), the random d-regular k-graph. By extending the switching technique of McKay and Wormald to k-graphs, we show that, for some range of d = d(n) and a constant c > 0, if m ~ cnd, then one can couple ℍ (k) (n,m) and ℍ (k) (n,d) so that the latter contains the former with probability tending to one as n → ∞. In view of known results on the existence of a loose Hamilton cycle in ℍ (k) (n,m), we conclude that ℍ (k) (n,d) contains a loose Hamilton cycle when d ≫ log n (or just dC log n, if k = 3) and d = o(n 1/2).

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Copyright
Copyright © Cambridge University Press 2014 

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References

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