Common neighbours in planar graphs

TL;DR

This paper provides a complete classification of planar graphs based on the number of common neighbors among vertex tuples, advancing the understanding of planar graphical degree sequences and resolving a significant open problem.

Contribution

It offers the first complete classification of planar graphs by common neighbor counts, addressing the longstanding problem of planar graphical n-degree sequences without multiplicities.

Findings

Classified all planar graphs by common neighbor counts for n-tuples.

Resolved the problem of planar graphical n-degree sequences without multiplicities.

Expanded the literature on planar graphical degree sequences.

Abstract

For every positive integer $n$, we find a complete classification for planar graphs according to the collection of numbers of common neighbours for every $n$-tuple of distinct vertices. Our results expand the literature on planar graphical degree sequences, that have recently been the object of renewed attention. Here we completely settle the version with no multiplicities of the vast problem of planar graphical $n$-degree sequences.