On planar bipartite biregular degree sequences

TL;DR

This paper characterizes when pairs of constant degree sequences are realizable as bipartite planar graphs, showing they are Eulerian except for two specific cases, thus advancing understanding of degree sequence realizability in planar bipartite graphs.

Contribution

It proves that constant degree sequence pairs are planar if and only if they are Eulerian, except for two specific non-planar cases, providing a complete characterization.

Findings

Constant pairs are planar iff they are Eulerian, excluding two specific cases.

Provides a complete classification of planar bipartite biregular degree sequences.

Identifies the exact non-planar pairs among constant 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 $(3^{25} \, | \, 5^{15})$.