Induced cycles in crossing graphs of median graphs

The crossing graph G # of a partial cube G has the equivalence classes of the Djoković–Winkler relation Θ as vertices, two Θ -classes being adjacent if they appear on some common isometric cycle. The following question [S. Klavžar, H.M. Mulder, Partial cubes and crossing graphs, SIAM J. Discrete Math. 15 (2002) 235–251, Problem 7.3] is treated: Let G be a median graph and n ≥ 4 . Does an induced cycle C n in G # necessarily force an induced cogwheel M n in G ? It is shown that the answer is positive for n = 4 , 5 and negative for n ≥ 6 . On the other hand it is proved that if G # contains an induced cycle C n , n ≥ 4 , then G contains some induced cogwheel M m , 4 ≤ m ≤ n . A refinement of the expansion procedure for partial cubes is obtained along the way.