  The biclique cover number (resp. biclique partition number) of a graph $G$,
$\mathrm{bc}(G$) (resp. $\mathrm{bp}(G)$), is the least number of biclique
(complete bipartite) subgraphs that are needed to cover (resp. partition) the
edges of $G$. The \emph{local biclique cover number} (resp. local biclique
partition number) of a graph $G$, $\mathrm{lbc}(G$) (resp. $\mathrm{lbp}(G)$),
is the least $r$ such that there is a cover (resp. partition) of the edges of
$G$ by bicliques with no vertex in more than $r$ of these bicliques. We show
that $\mathrm{bp}(G)$ may be bounded in terms of $\mathrm{bc}(G)$, in
particular, $\mathrm{bp}(G)\leq \frac{1}{2}(3^\mathrm{bc(G)}-1)$. However, the
analogous result does not hold for the local measures. Indeed, in our main
result, we show that $\mathrm{lbp}(G)$ can be arbitrarily large, even for
graphs with $\mathrm{lbc}(G)=2$. For such graphs, $G$, we try to bound
$\mathrm{lbp}(G)$ in terms of additional information about biclique covers of
$G$. We both answer and leave open questions related to this. There is a well
known link between biclique covers and subcube intersection graphs. We consider
the problem of finding the least $r(n)$ for which every graph on $n$ vertices
can be represented as a subcube intersection graph in which every subcube has
dimension $r$. We reduce this problem to the much studied question of finding
the least $d(n)$ such that every graph on $n$ vertices is the intersection
graph of subcubes of a $d$-dimensional cube.
