  The steady-state degree of a chemical reaction network is the number of
complex steady-states for generic rate constants and initial conditions. One
way to bound the steady-state degree is through the mixed volume of the
steady-state system or an equivalent system. In this work, we show that for
partionable binomial networks, whose resulting steady-state systems are given
by a set of binomials and a set of linear (not necessarily binomial)
conservation equations, computing the mixed volume is equivalent to finding the
volume of a single mixed cell that is the translate of a parallelotope. We then
turn our attention to identifying cycles with binomial steady-state ideals. To
this end, we give a coloring condition on directed cycles that guarantees the
network has a binomial steady-state ideal. We highlight both of these theorems
using a class of networks referred to as species-overlapping networks and give
a formula for the mixed volume of these networks.
