Maximizing Cliques in Shellable Clique Complexes

TL;DR

This paper explores how shellability in clique complexes influences the maximum number of cliques in graphs with bounded degree, extending classical extremal graph theory results.

Contribution

It characterizes graphs with shellable clique complexes that maximize cliques under degree constraints, linking topological properties to extremal graph theory.

Findings

Identifies graphs with maximum cliques given shellability constraints.

Establishes connections between shellability and clique maximization.

Extends Turán-type results to topologically constrained graph classes.

Abstract

In extremal graph theory, the problem of finding the elements of a given class of graphs which contain the most cliques traces its routes back to Tur\'an's famous theorem. We consider the implications of the connectivity property of simplicial complexes known as shellability on clique complexes associated with graphs. In this paper, we find the graphs which maximize cliques among all graphs on $n$ vertices with $\Delta(G)\leq r$ that have shellable clique complexes.