Cayley graphs for extraspecial p-groups and a covering graph perspective on Huang's theorem

TL;DR

This paper constructs new Cayley graphs for extraspecial p-groups, generalizing Cohen-Tits covers, and explores their connection to Huang's theorem and the Sensitivity Conjecture through a covering graph perspective.

Contribution

It introduces two infinite families of 4-cycle-free p-fold covers of Cartesian products of p-cycles using Cayley graphs of extraspecial p-groups, extending Cohen-Tits covers.

Findings

Constructed new Cayley graphs for extraspecial p-groups.

Generalized Cohen-Tits covers to infinite families.

Linked these graphs to Huang's theorem and the Sensitivity Conjecture.

Abstract

In 1985, Arjeh Cohen and Jacques Tits proved the existence of a 4-cycle-free 2-fold cover of the hypercube. This Cohen-Tits cover is closely related to the signed adjacency matrix that Hao Huang used last year in his proof of the Sensitivity Conjecture. Terence Tao observed that Huang's signed adjacency matrix can be understood by lifting functions on an elementary abelian 2-group to functions on a central extension. Inspired by Tao's observation, we generalize the Cohen-Tits cover by constructing, as Cayley graphs for extraspecial p-groups, two infinite families of 4-cycle-free p-fold covers of the Cartesian product of p-cycles.