Extendable shellability for $d$-dimensional complexes on $d+3$ vertices

TL;DR

This paper proves that all shellable $d$-dimensional complexes with up to $d+3$ vertices are extendably shellable, using graph theory and algebraic methods.

Contribution

It establishes extendable shellability for complexes with $d+3$ vertices, expanding understanding of shellability properties in combinatorial topology.

Findings

All such complexes are extendably shellable.

Connection between shellability and chordal graph structures.

Application of algebraic techniques to topological properties.

Abstract

We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection to linear quotients of quadratic monomial ideals.