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Common neighbours in planar graphs

Riccardo W. Maffucci

TL;DR

This work provides a complete classification of planar graphs according to the sets A_n(G) of possible numbers of common neighbours for n distinct vertices, for every positive integer n. The authors separate the analysis into an exceptional class 𝓔 for n≥3 and a parallel classification for n=2, detailing explicit patterns with parameters such as the maximum bipartite degree L and the maximum degree Δ. For n≥3, most planar graphs exhibit A_n(G) drawn from {0,1,2} in structured blocks, with detailed descriptions for polyhedra and outerplanar graphs; exceptional graphs 𝓣 and 𝓠 (and small special graphs) yield distinct A_n(G) patterns. For n=2, the paper achieves a thorough description using the exceptional class 𝓔′, including the D_ℓ and T′_ℓ families, and provides constructive methods to realise broad A_2 patterns. The results advance the study of planar graphical degree-like sequences and offer a unified framework connecting A_n classifications to classical degree sequences and polyhedral geometry, with potential extensions to other graph classes and n-degree problems.

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.

Common neighbours in planar graphs

TL;DR

This work provides a complete classification of planar graphs according to the sets A_n(G) of possible numbers of common neighbours for n distinct vertices, for every positive integer n. The authors separate the analysis into an exceptional class 𝓔 for n≥3 and a parallel classification for n=2, detailing explicit patterns with parameters such as the maximum bipartite degree L and the maximum degree Δ. For n≥3, most planar graphs exhibit A_n(G) drawn from {0,1,2} in structured blocks, with detailed descriptions for polyhedra and outerplanar graphs; exceptional graphs 𝓣 and 𝓠 (and small special graphs) yield distinct A_n(G) patterns. For n=2, the paper achieves a thorough description using the exceptional class 𝓔′, including the D_ℓ and T′_ℓ families, and provides constructive methods to realise broad A_2 patterns. The results advance the study of planar graphical degree-like sequences and offer a unified framework connecting A_n classifications to classical degree sequences and polyhedral geometry, with potential extensions to other graph classes and n-degree problems.

Abstract

For every positive integer , we find a complete classification for planar graphs according to the collection of numbers of common neighbours for every -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 -degree sequences.

Paper Structure

This paper contains 21 sections, 21 theorems, 107 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Common neighbours
  3. Common neighbours of three or more vertices
  4. Common neighbours of a pair of vertices
  5. Discussion
  6. Preliminaries.
  7. Plan of the paper.
  8. Proof of Theorems \ref{['thm:1']} and \ref{['thm:2']}
  9. The exceptional class $\mathcal{T}$
  10. Case of $\mathcal{T}_m$.
  11. Main arguments
  12. Case of $\mathcal{Q}_m$.
  13. Concluding the proof
  14. Proof of Theorem \ref{['thm:3']}
  15. $1\in A_2(G)$
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $G$ be a planar graph and $n\geq 3$ an integer. Then

Figures (10)

  • Figure 1: The families of polyhedra $T_m,B_\ell,B'_\ell$.
  • Figure 2: The exceptional graphs $S_5,S_7$.
  • Figure 3: The polyhedra $S_3,S_4,S_6,S_8,S_9$ and the planar, $2$-connected graph $S'$.
  • Figure 4: Lemma \ref{['le:no0']}.
  • Figure 5: If $G$ is planar and $0\not\in A_3(G)$, then in every plane immersion of $G$, every region is either triangular or quadrangular.
  • ...and 5 more figures

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof
  • Lemma 2.3
  • ...and 25 more