On planar bipartite biregular degree sequences

Abstract

A pair of sequences of natural numbers is called planar if there exists a simple, bipartite, planar graph for which the given sequences are the degree sequences of its parts. For a pair to be planar, the sums of the sequences have to be equal and Euler's inequality must be satisfied. Pairs that verify these two necessary conditions are called Eulerian. We prove that a pair of constant sequences is planar if and only if it is Eulerian (such pairs can be easily listed) and is different from $(3^5 \, | \, 3^5)$ and $(3^{25} \, | \, 5^{15})$.