Face 2-colorable quadrilateral embeddings of complete bipartite graphs

An embedding is said to be face 2-colorable if the faces of the embedding can be colored with two colors such that no two monochromatic faces share an edge. In this paper, it is proved that a face 2-colorable quadrilateral embedding of the complete bipartite graph K_m,n exists if and only if m and n are even. Moreover, we obtain a different proof of γ(K_m,n) = [ (m-2)(n-2)/4] which does not use rotational scheme and the methods known.