  A cubical polytope is a polytope with all its facets being combinatorially
equivalent to cubes. The paper is concerned with the linkedness of the graphs
of cubical polytopes. A graph with at least $2k$ vertices is $k$-linked if, for
every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of
vertices, there are $k$ vertex-disjoint paths joining the vertices in the
pairs.
  Larman and Mani in 1970 proved that simplicial $d$-polytopes, polytopes with
all their facets being combinatorially equivalent to simplices, are
$\floor{(d+1)/2}$-linked; this is the maximum possible linkedness given the
facts that a $\floor{(d+1)/2}$-linked graph is at least
$(2\floor{(d+1)/2}-1)$-connected and that some of these graphs are
$d$-connected but not $(d+1)$-connected.
  Here we establish that cubical $d$-polytopes are also
$\floor{(d+1)/2}$-linked for every $d\ne 3$; this is again the maximum possible
linkedness for such a class of polytopes.
