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 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube GG with girth more than 6 is a tree-zone graph and therefore 2n(G)−m(G)−i(G)+ce(G)=22n(G)−m(G)−i(G)+ce(G)=2 holds, where i(G)i(G) is the isometric dimension of GG and ce(G)ce(G) its convex excess.