Random partitions and edge-disjoint Hamiltonian cycles

• Authors: Stephen G. Hartke;Tyler Seacrest
• Venue: Journal of Combinatorial Theory Series B
• Year: 2013

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 @a(@d,n)=(@d+2@dn-n^2)/2. Christofides, Kuhn, and Osthus [3] proved that for every @e0, every graph G on a sufficiently large number n of vertices and minimum degree @d=n/2+@en contains @a(@d,n)/2-@en/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 @d=n/2+3n^3^/^4ln(n) contains @a(@d,n)/2-3n^7^/^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.