Two-ended regular median graphs

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs GG the condition of linear growth can be weakened to the condition that GG has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.