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Loose Hamilton Cycles in Regular Hypergraphs

Published online by Cambridge University Press:  24 September 2014

ANDRZEJ DUDEK ,
ALAN FRIEZE ,
ANDRZEJ RUCIŃSKI  and
MATAS ŠILEIKIS
ANDRZEJ DUDEK
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA (e-mail: [email protected])
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: [email protected])
ANDRZEJ RUCIŃSKI
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: [email protected])
MATAS ŠILEIKIS
Affiliation:
Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden (e-mail: [email protected])
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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 d ⩾ C log n, if k = 3) and d = o(n1/2).

Keywords

Primary 05C6505C80Secondary 05C38
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 179 - 194
Copyright
Copyright © Cambridge University Press 2014 

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