Linear Residuals and Gallai-Simplicial Complexes

TL;DR

This paper introduces a new algebraic criterion for shellability of simplicial complexes, explores Gallai-simplicial complexes of graphs, and demonstrates shellability in specific graph classes, advancing combinatorial topology.

Contribution

It provides a novel algebraic criterion for shellability and introduces Gallai-simplicial complexes, applying these concepts to various graph classes.

Findings

Doubly uni-cyclic graph's spanning simplicial complex is non-pure shellable.

Gallai-simplicial complexes can be analyzed for shellability using the new criterion.

The criterion applies to multiple classes of graphs, confirming shellability in those cases.

Abstract

In this paper, we give a new algebraic criterion for the {\em shellability} of (non-pure) simplicial complex $\Delta$ over $[n]$, shellable in the sense of Bj\"orner and Wachs \cite{BW}. We show that the spanning simplicial complex of doubly uni-cyclic graph is non-pure shellable. Moreover, we introduce the concept of Gallai-simplicial complex $\Delta_{\Gamma}(G)$ of a finite simple graph $G$. We applied the obtained criterion to discuss the shellability of Gallai simplicial complexes associated to various classes of graphs..