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Extendability of $1$-decomposable complexes

Rhea Ghosal, Melody Han, Benjamin Keller, Scarlett Kerr, Justin Liu, SuHo Oh, Ryan Tang, Chloe Weng

TL;DR

This work extends Simon's conjecture–level extendability to the broader class of pure $d$-dimensional $1$-decomposable complexes by proving a constructive, facet-by-facet extension to $oldsymbol{ riangle_{n+d-3}^{(d)}}$ that preserves $1$-decomposability. The authors develop a two-stage strategy: (i) extend to a fully coned complex over a new vertex set $H$ with $ig|Hig| ge d-2$, ensuring $1$-decomposability is maintained via shedding-face analysis and the gluing criterion; (ii) extend to the $d$-skeleton of the $(n+d-3)$-simplex using a high-dimensional analogue of the $2$-dimensional edge-extension, while controlling the skeleton via deletion and link arguments. This advances the understanding of extendability for a nontrivial intermediate class between vertex decomposable and shellable complexes and lays groundwork for future questions on removing extra vertices and generalizing to higher $k$-decomposability, moving closer to a full resolution of Simon's conjecture for broader families.

Abstract

A well-known conjecture of Simon (1994) states that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to $Δ_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex, by attaching one facet at a time while maintaining shellability. The notion of $k$-decomposability for simplicial complexes, which generalizes shellability, was introduced by Provan and Billera (1980). Coleman, Dochtermann, Geist, and Oh (2022) showed that any pure $d$-dimensional $0$-decomposable complex on $n$ vertices can similarly be extended to $Δ_{n-1}^{(d)}$, attaching one facet at a time while preserving $0$-decomposability. In this paper, we investigate the analogous question for $1$-decomposable complexes. We prove a slightly relaxed version: any pure $d$-dimensional $1$-decomposable complex on $n$ vertices can be extended to $Δ_{n + d - 3}^{(d)}$, attaching one facet at a time while maintaining $1$-decomposability.

Extendability of $1$-decomposable complexes

TL;DR

This work extends Simon's conjecture–level extendability to the broader class of pure -dimensional -decomposable complexes by proving a constructive, facet-by-facet extension to that preserves -decomposability. The authors develop a two-stage strategy: (i) extend to a fully coned complex over a new vertex set with , ensuring -decomposability is maintained via shedding-face analysis and the gluing criterion; (ii) extend to the -skeleton of the -simplex using a high-dimensional analogue of the -dimensional edge-extension, while controlling the skeleton via deletion and link arguments. This advances the understanding of extendability for a nontrivial intermediate class between vertex decomposable and shellable complexes and lays groundwork for future questions on removing extra vertices and generalizing to higher -decomposability, moving closer to a full resolution of Simon's conjecture for broader families.

Abstract

A well-known conjecture of Simon (1994) states that any pure -dimensional shellable complex on vertices can be extended to , the -skeleton of the -dimensional simplex, by attaching one facet at a time while maintaining shellability. The notion of -decomposability for simplicial complexes, which generalizes shellability, was introduced by Provan and Billera (1980). Coleman, Dochtermann, Geist, and Oh (2022) showed that any pure -dimensional -decomposable complex on vertices can similarly be extended to , attaching one facet at a time while preserving -decomposability. In this paper, we investigate the analogous question for -decomposable complexes. We prove a slightly relaxed version: any pure -dimensional -decomposable complex on vertices can be extended to , attaching one facet at a time while maintaining -decomposability.

Paper Structure

This paper contains 7 sections, 16 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic terminology
  4. Combinatorial tools
  5. Low-dimensional cases
  6. Main result
  7. Further questions

Key Result

Theorem 1.1

Given any pure $d$-dimensional $0$-decomposable simplicial complex $\mathcal{C}$ on $n$ vertices, we can extend $\mathcal{C}$ to $\Delta_{n-1}^{(d)}$ adding one facet at a time, while maintaining $0$-decomposability.

Theorems & Definitions (48)

  • Conjecture 1: Simon's Conjecture s94
  • Theorem 1.1: cdgo22
  • Definition 1
  • Definition 2
  • Definition 3: pb80
  • Example 1
  • Lemma 1: Gluing criterion
  • proof
  • Definition 4
  • Remark 1
  • ...and 38 more