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Complexes of graphs with bounded independence number

Minki Kim, Alan Lew

Abstract

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on the class of graphs with maximum degree bounded by $Δ$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $Δ$. Then, every collection of $\left\lfloor\left(\fracΔ{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$.

Complexes of graphs with bounded independence number

Abstract

Let be a graph and a positive integer. Let be the abstract simplicial complex whose simplices are the subsets of that do not contain an independent set of size in . We study the collapsibility numbers of the complexes for various classes of graphs, focusing on the class of graphs with maximum degree bounded by . As an application, we obtain the following result: Let be a claw-free graph with maximum degree at most . Then, every collection of independent sets in has a rainbow independent set of size .

Paper Structure

This paper contains 15 sections, 36 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes
  4. Homology and Leray numbers
  5. Graphs
  6. Upper bounds for collapsibility numbers
  7. Chordal graphs
  8. Graphs with bounded maximum degree
  9. The $n\leq 3$ case and claw-free graphs
  10. Rainbow independent sets in claw-free graphs
  11. Lower bounds on Leray numbers
  12. Extremal examples
  13. A counterexample to Conjecture \ref{['mainconj_collapsibility_version']}
  14. Leray number of the disjoint union of graphs
  15. Open problems

Key Result

Theorem 1.1

Let $X$ be a $d$-collapsible simplicial complex on vertex set $V$, and let $X^c=\{\sigma\subset V:\, \sigma\notin X\}.$ Then, every collection of $d+1$ sets in $X^c$ has a rainbow set belonging to $X^c$.

Theorems & Definitions (77)

  • Theorem 1.1: Kalai and Meshulam KM2005
  • Proposition 1.2
  • Conjecture 1.3: Aharoni, Briggs, Kim, Kim ABKK
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1: Wegner Weg75
  • ...and 67 more