Shellability in Clique-Free Complexes of Graphs

TL;DR

This paper investigates the shellability and algebraic properties of $t$-clique-free complexes derived from graphs, providing new conditions, constructions, and links to chordality and linear resolutions.

Contribution

It introduces new sufficient conditions for shellability of $t$-clique-free complexes and develops methods for constructing shellable complexes via clique attachments and whiskering.

Findings

$t$-clique-free complexes are shellable under certain graph conditions

Clique attachments preserve shellability and Cohen-Macaulayness

Chordal graphs' complements have $t$-linear resolutions

Abstract

We study combinatorial and algebraic properties of $t$-clique-free complexes, a family of simplicial complexes associated with finite simple graphs that generalize the classical independence complex. For a graph $G$ and an integer $t \ge 2$, the $t$-clique-free complex $\mathsf{CF}_t(G)$ is the simplicial complex on the vertex set of $G$ whose faces are the subsets inducing no cliques of size $t$. Our main results provide sufficient conditions for shellability and related decomposability properties of $t$-clique-free complexes. In particular, we show that if $G$ is a $t$-diamond-free chordal graph (in particular, a block graph), then $\mathsf{CF}_t(G)$ is $(t-2)$-decomposable and hence shellable. We also investigate how graph modifications via clique attachments influence shellability. Generalizing earlier constructions involving whiskers and clique extensions, we introduce the following operation: given a graph $H$, a subset $S \subseteq V(H)$, and an integer $t \ge 2$, we form a graph $\operatorname{Cl}(H,S,t)$ by attaching to each vertex in $S$ a clique of size at least $t$. We prove that $\mathsf{CF}_t(H\setminus S)$ is shellable if and only if $\mathsf{CF}_t(\operatorname{Cl}(H,S,t))$ is shellable. This yields a flexible method for constructing shellable complexes, particularly when $S$ is a cycle cover. In addition, we extend the notion of clique whiskering and show that for any graph admitting a clique vertex-partition, the resulting $t$-clique whiskering produces a pure and shellable, and hence Cohen-Macaulay, $t$-clique-free complex. Finally, we establish a Fr\"oberg-type result linking chordality and linear resolutions. We show that for any chordal graph $G$, the edge ideal of the complement $t$-clique clutter $\overline{\mathcal{CH}_t(G)}$ admits a $t$-linear resolution over any field.