  To any trivalent plane graph embedded in the sphere, Casals and Murphy
associate a differential graded algebra (dg-algebra), in which the underlying
graded algebra is free associative over a commutative ring. Our first result is
the construction of a generalization of the Casals--Murphy dg-algebra to
non-commutative coefficients, for which we prove various functoriality
properties not previously verified in the commutative setting. Our second
result is to prove that rank $r$ representations of this dg-algebra, over a
field $\mathbb{F}$, correspond to colorings of the faces of the graph by
elements of the Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$ so that
bordering faces are transverse, up to the natural action of
$\operatorname{PGL}_{2r}(\mathbb{F})$. Underlying the combinatorics, the
dg-algebra is a computation of the fully non-commutative Legendrian contact
dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not
prove as such in this paper. The graph coloring problem verifies that for
Legendrian 2-weaves, rank $r$ representations of the Legendrian contact
dg-algebra correspond to constructible sheaves of microlocal rank $r$. This is
the first explicit such computation of the bijection between the moduli spaces
of representations and sheaves for an infinite family of Legendrian surfaces.
