On packing Hamilton cycles in ε-regular graphs

Authors:
Alan Frieze;Michael Krivelevich
Affiliations:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA;Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Venue:
Journal of Combinatorial Theory Series B
Year:
2005

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Abstract

A graph G = (V, E) on n vertices is (α, ε)-regular if its minimal degree is at least αn, and for every pair of disjoint subsets S, T ⊂ V of cardinalities at least εn, the number of edges e(S, T) between S and T satisfies |e(S, T)/|S| |T|-α| ≤ ε. We prove that if α ≫ ε 0 are not too small, then every (α, ε)-regular graph on n vertices contains a family of (α/2 - O(ε))n edge-disjoint Hamilton cycles. As a consequence we derive that for every constant 0 p G(n, p), almost all edges can be packed into edge-disjoint Hamilton cycles. A similar result is proven for the directed case.