Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [21] proved that every graph with n vertices and minimum degree n/2 has at least b5n/224c edge-disjoint Hamiltonian cycles. In [20], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of b(n+ 4)/8c. Let α(δ, n) = (δ + √ 2δn− n2)/2. Christofides, Kühn, and Osthus [7] proved that for every > 0, every graph G on a sufficiently large number n of vertices and minimum degree δ ≥ n/2 + n contains α(δ, n)/2− n/4 edge-disjoint Hamiltonian cycles. Their proof uses Szemerédi’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 δ ≥ n/2 + 3n3/4 √ ln(n) contains α(δ, n)/2− 3n7/8(lnn)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 − √ min{deg(v), p} · ln(n) neighbors in each part.