Symmetry and optimality of disjoint Hamilton cycles in star graphs

Multiple edge-disjoint Hamilton cycles have been obtained in labelled star graphs Stn of degree n-1, using number-theoretic means, as images of a known base 2-labelled Hamilton cycle under label-mapping automorphisms of Stn. However, no optimum bounds for producing such edge-disjoint Hamilton cycles have been given, and no positive or negative results exist on whether Hamilton decompositions can be produced by such constructions other than a positive result for St5. We show that for all even n there exist such collections, here called symmetric collections, of φ(n)/2 edge-disjoint Hamilton cycles, where φ is Euler’s totient function, and that this bound cannot be improved for any even or odd n. Thus, Stn is not symmetrically Hamilton decomposable if n is not prime. Our method improves on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs.