On the sensitivity complexity of bipartite graph properties

Abstract

Sensitivity complexity, introduced by Cook, Dwork, and Reischuk (1982, 1986) in [2,3], is an important complexity measure of Boolean functions. Turán (1984) [7] initiated the study of sensitivity complexity for graph properties. He conjectured that for any non-trivial graph property on n vertices, the sensitivity complexity is at least n−1. He proved that it is greater than n/4 in his paper. Wegener (1985) [8] verified this conjecture for all monotone graph properties. Recently Sun (2011) [6] improved the lower bound to 617n for general graph properties. We follow their steps and investigate the sensitivity complexity of bipartite graph properties. In this paper we propose the following conjecture about the sensitivity of bipartite graph properties, which can be considered as the bipartite analogue of Turán’s conjecture: for any non-trivial n×m bipartite graph property f, s(f)≥max{⌈n+1m+1m⌉,⌈m+1n+1n⌉}. We prove this conjecture for all n×2 bipartite graph properties. For general n×m bipartite graph properties, we show a max{⌈n/2⌉,⌈m/2⌉} lower bound. We also prove this conjecture when the bipartite graph property can be written as a composite function.