Spurred primarily by a conjecture of Thurston in 1985, there was a series of developments in creating a "discrete analytic function" theory for maps between circle packings of complex domains. It seems that the main results in this area are discrete approximations of, and more constructive proofs of, classical complex analysis theorems (e.g. the Riemann mapping theorem by Rodin & Sullivan, Koebe uniformization by He & Schramm). Roughly speaking, are there results that go in the other direction, i.e. derive meaningful properties of discrete analytic functions from familiar combinatorial results?
For example, by the Four Color Theorem we know that the dual of a spherical triangulation is three-edge colorable, and we know that any triangulation admits a primal-dual circle packing (see, for example, Mohar & Thomassen 2001). Does this lead to anything meaningful about the discrete function theory of these primal-dual circle packings? Perhaps the answer is simply "no" given that the Four Color Theorem (and other combinatorial results) will be "global" statements instead of "local" ones.