  The Ollivier curvature has important applications in discrete geometry and
network theory, in particular as a measure of local clustering. The Ollivier
curvature is defined in terms of the Wasserstein distance which, in the
discrete setting, can be regarded as an optimal solution of a particular linear
programme. In certain classes of graph, this linear programme may be solved
\textit{a priori} giving rise to exact combinatorial expressions for the
Ollivier curvature. It has been claimed by Bhattacharya and Mukherjee (2013)
that an exact expression exists for the Ollivier curvature in bipartite graphs
and graphs of girth 5; we present counterexamples to these claims and identify
the error in the argument of Bhattacharya and Mukherjee. We then repeat the
analysis of Bhattacharya and Mukherjee for arbitrary graphs, taking this error
into account, and present reduced---parallelly solvable---linear programmes for
the calculation of the Ollivier curvature. This allows for potential
improvements in the exact numerical evaluation of the Ollivier curvature,
though the result heuristically suggests no general exact combinatorial
expression for the Ollivier curvature exists. Finally we give an exact
expression for the Ollivier curvature in a class of graphs defined by a
particular combinatorial constraint motivated by physical considerations.
