Face 2-colorable embeddings with faces of specified lengths

Suppose M=m1,m2,.,mr and N=n1,n2,.,nt are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2-colorable planar embedding in which the faces of one color have boundary lengths m1,m2,.,mr and the faces of the other color have boundary lengths n1,n2,.,nt. Such a graph is said to have a planar (M;N)-biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths m1,m2,.,mr and also into t cycles of lengths n1,n2,.,nt. Such a graph is said to be (M;N)-decomposable