Geodesics in non-positively curved plane tessellations

We introduce a natural combinatorial curvature function on the corners of plane tessellations and relate it to the global metric geometry of their corresponding edge and dual graphs. If the combinatorial curvature in the corners is non-positive then we prove that any geodesic path in such a graph may be extended to infinity. Moreover, if the combinatorial curvature is negative we show that every pair of geodesic segments with the same end points does not enclose any vertices. We apply these results to establish an estimate for the growth of distance balls, Gromov hyperbolicity, and four-colourability of certain classes of plane tessellations.