On Graphs and the Gotsman-Linial Conjecture for d = 2

TL;DR

This paper presents counterexamples to the Gotsman-Linial conjecture for quadratic functions and establishes an asymptotic form of the conjecture for specific graph classes, using elementary techniques.

Contribution

It provides the first infinite class of counterexamples for d=2 and proves an asymptotic version of the conjecture for certain graph-structured quadratic threshold functions.

Findings

Counterexamples to the Gotsman-Linial conjecture for d=2

Asymptotic form of the conjecture for low fractional chromatic number graphs

Elementary and self-contained proof techniques

Abstract

We give an infinite class of counterexamples to the Gotsman-Linial conjecture when d = 2. On the other hand, we establish an asymptotic form of the conjecture for quadratic threshold functions whose non-zero quadratic terms define a graph with either low fractional chromatic number or few edges. Our techniques are elementary and our exposition is self-contained, if you're into that.