## DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 363-368

## 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

### DMTCS Conference Volume AE (2005), pp. 363-368

author: Tomáš Dvořák, Petr Gregor and Václav Koubek Spanning paths in hypercubes Hamiltonian paths, spanning paths, hypercube, vertex fault tolerance Given a family {u i ,v i } i=1 k of pairwise distinct vertices of the n -dimensional hypercube Q n such that the distance of u i and v i is odd and k≤n-1 , there exists a family {P i } i=1 k of paths such that u i and v i are the endvertices of P i and {V(P i )} i=1 k partitions V(Q n ) . This holds for any n≥2 with one exception in the case when n=k+1=4 . On the other hand, for any n≥3 there exist n pairs of vertices satisfying the above condition for which such a family of spanning paths does not exist. We suggest further generalization of this result and explore a relationship to the problem of hamiltonicity of hypercubes with faulty vertices. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Tomáš Dvořák and Petr Gregor and Václav Koubek (2005), Spanning paths in hypercubes, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 363-368 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAE0170.ps.gz (66 K) dmAE0170.ps (190 K) dmAE0170.pdf (139 K)

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