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Indivisibility for Classes of Graphs

Vince Guingona, Felix Nusbaum, Zain Padamsee, Miriam Parnes, Christian Pippin, Ava Zinman

TL;DR

This work classifies indivisibility for significant graph classes, revealing a sharp threshold for the heredity-based sparsity classes: the family of hereditarily \\alpha-sparse graphs is indivisible precisely when \\alpha > 2$, with a parallel threshold \\alpha \ge 2$ for the strengthened class \\mathbf{K}_\alpha^+. The paper also proves indivisibility for cographs, perfect graphs, and chordal graphs, while showing non-indivisibility for threshold, split, and distance-hereditary graphs. It situates these results within a framework that leverages the Amalgamation Property and forbidding induced substructures, and it analyzes when amalgamation is present and how it relates to indivisibility. Together, these results deepen the Ramsey-theoretic understanding of graph classes and point to new lines of inquiry on generalized substructure notions and finite-color Ramsey degrees.

Abstract

We examine indivisibility for classes of graphs. We show that the class of hereditarily $α$-sparse graphs is indivisible if and only if $α> 2$. Additionally, we show that the following classes of graphs are indivisible: perfect graphs, cographs, and chordal graphs, and the following classes of graphs are not indivisible: threshold graphs, split graphs, and distance-hereditary graphs.

Indivisibility for Classes of Graphs

TL;DR

This work classifies indivisibility for significant graph classes, revealing a sharp threshold for the heredity-based sparsity classes: the family of hereditarily \\alpha-sparse graphs is indivisible precisely when \\alpha > 2 for the strengthened class \\mathbf{K}_\alpha^+. The paper also proves indivisibility for cographs, perfect graphs, and chordal graphs, while showing non-indivisibility for threshold, split, and distance-hereditary graphs. It situates these results within a framework that leverages the Amalgamation Property and forbidding induced substructures, and it analyzes when amalgamation is present and how it relates to indivisibility. Together, these results deepen the Ramsey-theoretic understanding of graph classes and point to new lines of inquiry on generalized substructure notions and finite-color Ramsey degrees.

Abstract

We examine indivisibility for classes of graphs. We show that the class of hereditarily -sparse graphs is indivisible if and only if . Additionally, we show that the following classes of graphs are indivisible: perfect graphs, cographs, and chordal graphs, and the following classes of graphs are not indivisible: threshold graphs, split graphs, and distance-hereditary graphs.
Paper Structure (6 sections, 38 theorems, 45 equations)

This paper contains 6 sections, 38 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hereditarily sparse graphs
  4. Classes of graphs with forbidden substructures
  5. The Amalgamation Property
  6. Future Directions

Key Result

Theorem 2.4

A class $\mathbf{K}$ is indivisible if and only if, for all $A \in \mathbf{K}$, there exists $B \in \mathbf{K}$ such that, for all $2$-colorings $c$ of $B$, $B$ has a monochromatic copy of $A$ with respect to $c$.

Theorems & Definitions (73)

  • Definition 2.1: Hereditary Property
  • Definition 2.2: Amalgamation Property
  • Definition 2.3: Indivisible
  • Theorem 2.4: Theorem 1 of EZS91
  • Definition 2.5: Boundary
  • Theorem 2.6: Sau20
  • Lemma 2.7
  • Definition 2.8: $\mathbf{K}_\alpha$ and $\mathbf{K}_\alpha^+$
  • Remark 2.9
  • Lemma 2.10
  • ...and 63 more