Random partitions and edge-disjoint Hamiltonian cycles

Abstract

Nash-Williams [1] proved that every graph with n vertices and minimum degree n/2 has at least ⌊5n/224⌋ 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 ⌊(n+4)/8⌋.Let α(δ,n)=(δ+2δn−n2)/2. Christofides, Kühn, and Osthus [3] proved that for every ϵ>0, every graph G on a sufficiently large number n of vertices and minimum degree δ⩾n/2+ϵn contains α(δ,n)/2−ϵn/4 edge-disjoint Hamiltonian cycles. Their proof uses Szemerédiʼs 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+3n3/4ln(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.