A new proof of Vلzsonyiʹs conjecture

The diameter graph G of n points in Euclidean 3-space has a bipartite, centrally symmetric double covering on the sphere. Three easy corollaries follow: (1) A self-contained proof of Vázsonyiʹs conjecture that G has at most 2 n − 2 edges, which avoids the ball polytopes used in the original proofs given by Grünbaum, Heppes and Straszewicz. (2) G can be embedded in the projective plane. (3) Any two odd cycles in G intersect [V.L. Dolʹnikov, Some properties of graphs of diameters, Discrete Comput. Geom. 24 (2000) 293–299].