Face 2-colorable quadrilateral embeddings of complete bipartite graphs

Abstract

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,K-n. exists if and only if m and n are even. Moreover, we obtain a different proof of gamma(K-m,K-n) = [(m-2)(n-2)/4] which does not use rota4 tional scheme and the methods known.