On partial cubes and graphs with convex intervals

摘要: A graph is called a partial cube if it admits an isometric embedding into hypercube. Subdivisions of wheels are considered with respect to such embeddings and the convexity their intervals. This allows us answer in negative question Chepoi Tardif from 1994 whether all bipartite graphs convex intervals cubes. On positive side we prove that which bipartite, has intervals, not cube, always contains subdivision $K_4$.