Counterexamples to "A Conjecture on Induced Subgraphs of Cayley Graphs" [arXiv:2003.13166]

TL;DR

This paper disproves a conjecture that all Cayley graphs possess a property related to induced subgraphs with high degree, by providing counterexamples including an infinite family with low maximum degree on large vertex sets.

Contribution

It constructs counterexamples to the conjecture that all Cayley graphs have the property, including an infinite family with unbounded valency and low maximum degree on large subsets.

Findings

Counterexamples to the conjecture are constructed.

An infinite family of Cayley graphs with unbounded valency is provided.

These graphs have large subsets with low maximum degree.

Abstract

Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$, where $d$ is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian $2$-group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove this conjecture by constructing various counterexamples, including an infinite family of Cayley graphs of unbounded valency which admit an induced subgraph of maximum valency $1$ on a set of more than half its vertices.