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Halin graphs with positive Lin-Lu-Yau curvature

Kaizhe Chen, Huiqiu Lin, Shiping Liu, Zhe You

TL;DR

The paper addresses the problem of characterizing generalized Halin graphs that have positive Lin-Lu-Yau curvature. It develops a case-based analysis grounded in the Laplacian formulation of $\kappa_{LLY}$ and transport-geometry tools to constrain the structure of the generating tree $T$ and its leaves, yielding a finite classification. The main contributions are a sharp bound $|V(G)|\le 12$ for generalized Halin graphs with positive $\kappa_{LLY}$ and an explicit list of realizations: $W_n$ ($4\le n\le 12$), $W'_n$ ($5\le n\le 9$), $W''_n$ ($6\le n\le 10$), and $H_i$ ($1\le i\le 8$), with Halin graphs narrowing to $W_n$ ($4\le n\le 12$) and $H_i$ ($i=3,7$). This provides a complete curvature-based classification for a natural class of planar graphs and offers tools for recognizing positive-curvature structures in graph theory. The results advance our understanding of how discrete curvature constraints shape planar graph families and have potential implications for graph algorithms and geometric analysis on graphs.

Abstract

Halin graphs constitute an interesting class of planar and polyhedral graphs. A generalized Halin graph is obtained by connecting all leaves of a planar embedding of a tree via a cycle. A Halin graph is a generalized Halin graph having no vertex of degree two. We classify all generalized Halin graphs with positive Lin-Lu-Yau curvature.

Halin graphs with positive Lin-Lu-Yau curvature

TL;DR

The paper addresses the problem of characterizing generalized Halin graphs that have positive Lin-Lu-Yau curvature. It develops a case-based analysis grounded in the Laplacian formulation of and transport-geometry tools to constrain the structure of the generating tree and its leaves, yielding a finite classification. The main contributions are a sharp bound for generalized Halin graphs with positive and an explicit list of realizations: (), (), (), and (), with Halin graphs narrowing to () and (). This provides a complete curvature-based classification for a natural class of planar graphs and offers tools for recognizing positive-curvature structures in graph theory. The results advance our understanding of how discrete curvature constraints shape planar graph families and have potential implications for graph algorithms and geometric analysis on graphs.

Abstract

Halin graphs constitute an interesting class of planar and polyhedral graphs. A generalized Halin graph is obtained by connecting all leaves of a planar embedding of a tree via a cycle. A Halin graph is a generalized Halin graph having no vertex of degree two. We classify all generalized Halin graphs with positive Lin-Lu-Yau curvature.
Paper Structure (7 sections, 9 theorems, 27 equations, 5 figures)

This paper contains 7 sections, 9 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ollivier's Ricci curvature and Lin-Lu-Yau curvature
  4. Generalized Halin graphs
  5. Proof of Theorem \ref{['generalized Halin graph with positive LLY']}
  6. An example
  7. Appendix

Key Result

Theorem 1.1

Let $G=(V,E)$ be a connected planar graph such that every vertex has degree at least $3$. If the Lin-Lu-Yau curvature is positive at every edge, then $G$ is finite. In particular, $|V (G)| \leq 17^{544}$.

Figures (5)

  • Figure 1: Generalized Halin graphs $H_i, 1\leq i\leq 8$
  • Figure 2: A example of labelling outer vertices in each component of $T-\{x\}$
  • Figure 3: Graph $H'$
  • Figure 4: Remaining graphs in Subcase 1.2
  • Figure 5: Remaining graphs in Subcase 2.2

Theorems & Definitions (17)

  • Theorem 1.1: Lu-Wang LW20
  • Theorem 1.2: Liu-Lu-Wang LLW24
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: Wasserstein distance
  • Definition 2.2: Lin-Lu-Yau curvature
  • Theorem 2.3: Curvature via the Laplacian MW19
  • Lemma 2.4: LLY14
  • proof
  • Definition 2.5
  • ...and 7 more