On the Sensitivity Complexity of k-Uniform Hypergraph Properties

Abstract

In this article, we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n(⌈k/3⌉) for any k≥3, where n is the number of vertices. Moreover, we can do better when k≡1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n⌈k/3⌉-1/2). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Ω (nk/2). We also investigate the sensitivity complexity of other symmetric functions and show that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O(N1/3), where N is the number of variables.