Abstract
We discuss the existence of vertex disjoint path coverings with prescribed ends for the \(n\)-dimensional hypercube with or without deleted vertices. Depending on the type of the set of deleted vertices and desired properties of the path coverings we establish the minimal integer \(m\) such that for every \(n \ge m\) such path coverings exist. Using some of these results, for \(k \le 4\), we prove Locke’s conjecture that a hypercube with \(k\) deleted vertices of each parity is Hamiltonian if \(n \ge k +2.\) Some of our lemmas substantially generalize known results of I. Havel and T. Dvořák. At the end of the paper we formulate some conjectures supported by our results.
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