Kruskal--Katona type theorems for clique complexes arising from chordal and strongly chordal graphs

TL;DR

This paper extends Kruskal--Katona theorems to clique complexes of chordal and strongly chordal graphs, characterizing their combinatorial properties and shellability conditions.

Contribution

It introduces Kruskal--Katona type theorems for forests and quasi-forests, and characterizes shellability of quasi-forests via their h-vectors.

Findings

Kruskal--Katona theorems for forests and quasi-forests

Shellability of quasi-forests characterized by h-vector conditions

Structural properties of clique complexes from chordal graphs

Abstract

A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. In addition, it will be shown that a quasi-forest is shellable if and only if its $h$-vector $(h_0, h_1, h_2, ...)$ satisfies $h_i = 0$ for $i > 1$.