On cube-free median graphs

Let G be a cube-free median graph. It is proved that k/2>=n-1>=m/2n>=s>=r-1, where n, m, s, k, and r are the number of vertices, edges, squares, @Q-classes, and the number of edges in a smallest @Q-class of G, respectively. Moreover, the equalities characterize Cartesian products of two trees of the same order. The cube polynomial of cube-free median graphs is also considered and it is shown that planar cube-free median graphs can be recognized in linear time.