Partial Cubes and Crossing Graphs

Partial cubes are defined as isometric subgraphs of hypercubes. For a partial cube G, its crossing graph G# is introduced as the graph whose vertices are the equivalence classes of the Djoković–Winkler relation Θ, two vertices being adjacent if they cross on a common cycle. It is shown that every graph is the crossing graph of some median graph and that a partial cube G is 2-connected if and only if G# is connected. A partial cube G has a triangle-free crossing graph if and only if G is a cube-free median graph. This result is used to characterize the partial cubes having a tree or a forest as its crossing graph. An expansion theorem is given for the partial cubes with complete crossing graphs. Cartesian products are also considered. In particular, it is proved that G# is a complete bipartite graph if and only if G is the Cartesian product of two trees.