Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [1] proved that every graph with n vertices and minimum degree n/2 has at least left perpendicular5n/224right perpendicular edge-disjoint Hamiltonian cycles. In [2], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of left perpendicular(n + 4)/8right perpendicular. Let alpha(delta, n)= (delta + root 2 delta n-n(2))/2. Christofides, Kuhn, and Osthus [3] proved that for every epsilon > 0, every graph G on a sufficiently large number n of vertices and minimum degree delta >= n/2 + epsilon n contains alpha(delta, n)/2 - epsilon n/4 edge-disjoint Hamiltonian cycles. Their proof uses Szemeredi's Regularity Lemma, and hence the "sufficiently large" requirement on n is a strong condition. In this paper we prove a similar result using methods that do not rely on the Regularity Lemma. In particular, we prove that every graph on n vertices with minimum degree delta >= n/2 + 3n(3/4)root ln(n) contains alpha(delta, n)/2 - 3n(7/8)(ln n)(1/4)/2 edge-disjoint Hamiltonian cycles. Our proof rests on a structural result that is of independent interest: let G be a graph on n vertices, where n = pq. Then there exists a partition of the vertices of G into q parts of size p such that every vertex v has at least deg(v)/q - root min[deg(v), p} . ln(n) neighbors in each part. (C) 2013 Elsevier Inc. All rights reserved.