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Path coverings with prescribed ends in faulty hypercubes
We discuss the existence of vertex disjoint path coverings with prescribed
ends for the -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 such that for every such path coverings exist. Using some of these results, for ,
we prove Locke's conjecture that a hypercube with deleted vertices of each
parity is Hamiltonian if Some of our lemmas substantially
generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the
paper we formulate some conjectures supported by our results.Comment: 26 page
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