Crossing Graphs as Joins of Graphs and Cartesian Products of Median Graphs

For a partial cube G its crossing graph G is the graph with vertices representing Θ-classes of G, two classes being adjacent if they cross on some cycle in G. The following problem posed in [11, Problem 7.1] is considered: what can be said about the partial cube G if G is the join A⊕B of not edge-less graphs A and B? It is proved that for arbitrary graphs A and B, where at least one of them contains an edge, there exists a Cartesian prime partial cube G such that G = A⊕B. On the other hand, if G is a median graph, then G = A ⊕ B if and only if G = H K, where H = A and K = B. Along the way some new facts about partial cubes are obtained. 2000 Mathematical Subject Classification: 05C75, 05C12.