A note on simplicial cliques

TL;DR

This paper proves the existence of simplicial cliques in certain classes of graphs, motivated by applications in physics and quantum information, extending known results to more general graph classes.

Contribution

It establishes the existence of simplicial cliques in a broader class of graphs defined by forbidden induced subgraphs, generalizing previous results.

Findings

Every non-null even-hole-free claw-free graph has a simplicial clique.

The existence of simplicial cliques is proven in more general graph classes.

Results are motivated by applications in physics and quantum information.

Abstract

Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique $K$ such that for every vertex $v \in K$, the set of neighbours of $v$ outside of $K$ is a clique. In fact, we prove the existence of a simplicial clique in a more general class of graphs defined by forbidden induced subgraphs.