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arXiv · 2508.04555

Extendability of $1$-decomposable complexes

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

A well-known conjecture of Simon (1994) states that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to $\Delta_{n-1}^{(d)}$, 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 $\Delta_{n-1}^{(d)}$, 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 $\Delta_{n + d - 3}^{(d)}$, attaching one facet at a time while maintaining $1$-decomposability.

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Rhea Ghosal, Melody Han, Benjamin Keller, Scarlett Kerr, Justin Liu, SuHo Oh, Ryan Tang, Chloe Weng. 2025-08-06. Extendability of $1$-decomposable complexes. https://arxiv.org/abs/2508.04555

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