A Conjecture on Induced Subgraphs of Cayley Graphs

Abstract: In this paper, we propose the following conjecture which generalizes a theorem proved by Huang [Hua19] in his recent breakthrough proof of the sensitivity conjecture. We conjecture that for any Cayley graph $X = \Gamma(G,S)$ on a group $G$ and any generating set $S$, if $U \subseteq G$ has size $|U| > |G|/2$, then the induced subgraph of $X$ on $U$ has maximum degree at least $\sqrt{|S|/2}$. Using a recent idea of Alon and Zheng [AZ20], who proved this conjecture for the special case when $G = Z_2^n$, we prove that this conjecture is true whenever $G$ is abelian. We also observe that for this conjecture to hold for a graph $X$, some symmetry is required: it is insufficient for $X$ to just be regular and bipartite.