There are no finite partial cubes of girth more than 6 and minimum degree at least 3

Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than six must have vertices of degree less than three. As a direct corollary we get that every regular partial cube of girth more than six is an even cycle. Along the way we prove that every partial cube G with girth more than six is the so-called zone graph and therefore 2n(G) − m(G) − i(G) + ce(G) = 2 holds, where i(G) is the isometric dimension of G and ce(G) its convex excess.