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On the matching complexes of categorical product of path graphs

Raju Kumar Gupta, Sourav Sarkar, Sagar S. Sawant, Samir Shukla

Abstract

The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appears in various places and found applications in many areas of mathematics including; discrete geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of categorical product $P_n \times P_m$ of path graphs $P_n$ and $P_m$. For $m = 1$, $P_n \times P_m$ is a discrete graph and therefore its matching complex is the void complex. For $m = 2$, $\mathsf{M}(P_n \times P_m)$ has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for $n \geq 2$ and $3 \leq m \leq 5$, the matching complex of $P_n \times P_m$ is homotopy equivalent to a wedge of spheres. For $m =3$, we give a closed form formula for the number and dimension of spheres appearing in the wedge. Further, for $m \in \{4, 5\}$, we give minimum and maximum dimension of spheres appearing in the wedge in the homotopy type of $\mathsf{M}(P_n \times P_m)$.

On the matching complexes of categorical product of path graphs

Abstract

The matching complex of a graph is a simplicial complex whose simplices are matchings in . These complexes appears in various places and found applications in many areas of mathematics including; discrete geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of categorical product of path graphs and . For , is a discrete graph and therefore its matching complex is the void complex. For , has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for and , the matching complex of is homotopy equivalent to a wedge of spheres. For , we give a closed form formula for the number and dimension of spheres appearing in the wedge. Further, for , we give minimum and maximum dimension of spheres appearing in the wedge in the homotopy type of .
Paper Structure (22 sections, 19 theorems, 109 equations, 31 figures, 2 tables)

This paper contains 22 sections, 19 theorems, 109 equations, 31 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matching complexes of Matching of grid
  4. Independence complexes of $\Gamma_{n,5}$ and $\Lambda_{n,5}$
  5. Definitions of intermediary graphs and their independence complexes for $n = 0$
  6. General case computation
  7. Independence complex of Ind(Lambda n, for n >=0
  8. Homotopy type of $\mathsf{M}(P_n \times P_5)$
  9. Matching complexes of Pn x P4
  10. Graph $\Gamma_{n,4}$
  11. Graph $\Lambda_{n,4}$
  12. Graph $A_{n,4}$
  13. Graph $B_{n,4}$
  14. Graph $C_{n,4}$
  15. Graph $D_{n,4}$
  16. ...and 7 more sections

Key Result

Theorem 1.3

For $n \geq 2$ and $3 \leq m \leq 5$, $\mathsf{M}(P_n \times P_m)$ is homotopy equivalent to a wedge of spheres.

Figures (31)

  • Figure 1: Cartesian and categorical product of $P_7$ and $P_5$
  • Figure 2: $\Gamma_{n,5}$
  • Figure 3: $\Lambda_{n, 5}$
  • Figure 4: $\Tilde{\Gamma}_{n,5}$
  • Figure 5: $A_{n,5}$
  • ...and 26 more figures

Theorems & Definitions (85)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 2.1
  • Proposition 2.2: Proposition 3.1, Adamaszek
  • Proposition 2.3: Lemma 2.4, Engstrom
  • Proposition 2.4: Lemma 2.5, Engstrom
  • Proposition 2.5: Proposition 3.4, Adamaszek
  • ...and 75 more