Locally 4-by-4 grid graphs

We investigate locally grid graphs. The main results are (i) a characterization of the Johnson graphs (and certain quotients of these) as locally grid graphs such that two points at distance 2 have precisely four common neighbors, and (ii) a complete determination of all graphs that are locally a 4 x 4 grid (it turns out that there are four such graphs, with 35, 40, 40, and 70 vertices).