Spanning Edge Cyclability of Enhanced Hypercube Networks

The Hamiltonian problem is a fundamental research topic in graph theory. In this paper, we study a relaxation of the Hamiltonian problem. A graph [Formula: see text] is spanning [Formula: see text]-edge-cyclable if, for any [Formula: see text] independent edges [Formula: see text] of [Formula: see text], there exist [Formula: see text] vertex-disjoint cycles [Formula: see text] in [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]. Clearly, a graph [Formula: see text] is Hamiltonian if it is spanning [Formula: see text]-edge-cyclable. We focus on spanning [Formula: see text]-edge-cyclability of enhanced hypercube network, which is an important variant of well known hypercube. We prove that the [Formula: see text]-dimensional enhanced hypercube [Formula: see text] is spanning [Formula: see text]-edge-cyclable for [Formula: see text], [Formula: see text] and odd [Formula: see text].