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Large expander subgraphs in high genus triangulations

Tanguy Lions, Baptiste Louf

Abstract

We prove that random triangulations of high genus contain very large expander subgraphs, answering a question of Benjamini. Our approach relies on new general criteria for arbitrary graphs to contain large expander subgraphs.

Large expander subgraphs in high genus triangulations

Abstract

We prove that random triangulations of high genus contain very large expander subgraphs, answering a question of Benjamini. Our approach relies on new general criteria for arbitrary graphs to contain large expander subgraphs.
Paper Structure (7 sections, 9 theorems, 37 equations, 5 figures)

This paper contains 7 sections, 9 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Definitions
  3. Graphs
  4. Maps and triangulations
  5. Proof of the main result: \ref{['thm_triangulations']}
  6. From isolated vertices to expander subgraphs: proof of \ref{['thm_isolated_to_expander']}
  7. Expander subgraphs and duality: proof of \ref{['thm_dual']}

Key Result

Theorem 1

Fix a sequence $(g_n)_{n\geq0}$ such that $\frac{g_n}{n}\to \theta \in(0,1/2)$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ on $3n$ edges, and $\mathbf{T}_{2n,g_n}^*$ its dual map. Then for every $\varepsilon>0$, there exists $\kappa>0$ depending only on $\theta$ and $\vareps

Figures (5)

  • Figure 1: On this example, we represent a graph $G$ and $X \subset G$ connected that is a $\kappa$-bad set with $\kappa = \frac{1}{10}$. It is also a strong $\kappa$-bad set.
  • Figure 2: On this example, we represent on top the subset $X \subset G$ and on the bottom we represent the connected components $C_1,\cdots,C_r$ of $G \setminus X$.
  • Figure 3: On the top: the subgraph $G_t \subset G$. On the bottom: the connected components $S_1,\cdots,S_t$ that have been removed by the process. The subset $S_{\mathcal{J}}$ is the blue part.
  • Figure 4: On the left: the map $M$ is represented in black. The subgraph $G^{*}$ of $M^{*}$ is represented in red. The subgraph $G$ of $M$ is represented in blue. On the right: the subgraph $X$ of $G$ is represented in green. The subgraph $X^{*}$ of $G^{*}$ is represented in orange.
  • Figure 5: We represent in purple the set $F \subset X^{*}$ in purple. Note that it exactly corresponds to the set of vertices in $X^{*}$ that are incident to a red edge.

Theorems & Definitions (21)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • proof
  • ...and 11 more