Optimal Bounds for Disjoint Hamilton Cycles in Star Graphs

In interconnection network topologies, the [Formula: see text]-dimensional star graph [Formula: see text] has [Formula: see text] vertices corresponding to permutations [Formula: see text] of [Formula: see text] symbols [Formula: see text] and edges which exchange the positions of the first symbol [Formula: see text] with any one of the other symbols. The star graph compares favorably with the familiar [Formula: see text]-cube on degree, diameter and a number of other parameters. A desirable property which has not been fully evaluated in star graphs is the presence of multiple edge-disjoint Hamilton cycles which are important for fault-tolerance. The only known method for producing multiple edge-disjoint Hamilton cycles in [Formula: see text] has been to label the edges in a certain way and then take images of a known base 2-labelled Hamilton cycle under different automorphisms that map labels consistently. However, optimal bounds for producing edge-disjoint Hamilton cycles in this way, and whether Hamilton decompositions can be produced, are not known for any [Formula: see text] other than for the case of [Formula: see text] which does provide a Hamilton decomposition. In this paper we show that, for all n, not more than [Formula: see text], where [Formula: see text] is Euler’s totient function, edge-disjoint Hamilton cycles can be produced by such automorphisms. Thus, for non-prime [Formula: see text], a Hamilton decomposition cannot be produced. We show that the [Formula: see text] upper bound can be achieved for all even [Formula: see text]. In particular, if [Formula: see text] is a power of 2, [Formula: see text] has a Hamilton decomposable spanning subgraph comprising more than half of the edges of [Formula: see text]. Our results produce a better than twofold improvement on the known bounds for any kind of edge-disjoint Hamilton cycles in [Formula: see text]-dimensional star graphs for general [Formula: see text].

Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ=(1/2+α)n. For any constant α>0, we give an optimal answer in the following sense: let regeven(n,δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals regeven(n,δ)/2. The value of regeven(n,δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.