2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 363368
author:  Tomáš Dvořák, Petr Gregor and Václav Koubek 

title:  Spanning paths in hypercubes 
keywords:  Hamiltonian paths, spanning paths, hypercube, vertex fault tolerance 
abstract: 
Given a family
{u
of pairwise distinct vertices of the
i
,v
i
}
i=1
k
n
dimensional hypercube
Q
such that the distance of
n
u
and
i
v
is odd and
i
k≤n1
, there exists a family
{P
of paths such that
i
}
i=1
k
u
and
i
v
are the endvertices of
i
P
and
i
{V(P
partitions
i
)}
i=1
k
V(Q
. This holds for any
n
)
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.

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reference:  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. 363368 
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