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Box complexes: at the crossroad of graph theory and topology

Hamid Reza Daneshpajouh, Frédéric Meunier

Abstract

Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their topological properties. They provide thus a fascinating topic mixing topology and discrete mathematics. This paper is intended to provide an up-do-date survey on box complexes. It is based on classical results and recent findings from the literature, but also establishes new results improving our current understanding of the topic, and identifies several challenging open questions.

Box complexes: at the crossroad of graph theory and topology

Abstract

Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their topological properties. They provide thus a fascinating topic mixing topology and discrete mathematics. This paper is intended to provide an up-do-date survey on box complexes. It is based on classical results and recent findings from the literature, but also establishes new results improving our current understanding of the topic, and identifies several challenging open questions.
Paper Structure (28 sections, 23 theorems, 25 equations, 4 figures)

This paper contains 28 sections, 23 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Box complexes: definition, relevance, and main properties
  3. Definition
  4. Chromatic number and connectivity
  5. "Universality" of the box complex
  6. Further results on box complexes and other relevant complexes
  7. Another box complex
  8. Other complexes
  9. Box complex of some specific graphs
  10. Parameters of graphs and complexes
  11. Combinatorial parameters
  12. Topological parameters
  13. General relations between various parameters
  14. Computability and complexity of parameters
  15. Decidability and hardness
  16. ...and 13 more sections

Key Result

Theorem 2.1

Every free simplicial $\mathbb{Z}_2$-complex is $\mathbb{Z}_2$-homotopy equivalent to the box complex $\mathsf{B}(G)$ of some graph $G$.

Figures (4)

  • Figure 1: Each arrow represents a $\leqslant$ inequality, from the smaller parameter to the larger. Apart from the clique number, the parameters below the "Borsuk--Ulam boundary" requires the Borsuk--Ulam theorem---or a "stronger" statement---to be established as a lower bound on the chromatic number. This is further discussed in Section \ref{['sec:comput']}.
  • Figure 2: The $6$-cycle and its box complex
  • Figure 3: Illustration of Csorba's construction underlying the proof of Theorem \ref{['thm:csorba']}
  • Figure 4: The $4$-cycle and its box complex $\mathsf{B}_0(C_4)$

Theorems & Definitions (71)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 2.1: Csorba csorba2007homotopy
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Theorem 3.1: Csorba csorba2007homotopy
  • ...and 61 more