A characterization of cocircuit graphs of uniform oriented matroids

The cocircuit graph of an oriented matroid is the 1-skeleton of the cellular decomposition induced by the Topological Representation Theorem due to Folkman and Lawrence (1978) [J. Folkman, J. Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978) 199–236]. In this paper we exhibit a characterization of such graphs (for the uniform case) via their natural embedding into Q n k —the 1-skeleton of the n-cubeʹs k-skeletonʹs dual complex. The main theorem reads, basically, as follows: A graph G is the cocircuit graph of a d-dimensional uniform oriented matroid on n elements if and only if its order is 2 ( n d + 2 ) , and it can be embedded antipodally and “metrically” into Q n n − d − 2 .