Around the positive graph conjecture

TL;DR

This paper advances understanding of the positive graph conjecture by proving structural properties of positive graphs, identifying counterexamples to hypergraph analogues, and linking positivity to graph codes.

Contribution

It proves that connected positive graphs must have an even degree vertex, finds new counterexamples to hypergraph Sidorenko-type conjectures, and connects positive graphs to vanishing graph code density.

Findings

Connected positive graphs have a vertex of even degree

r-uniform tight cycles for odd r are counterexamples to hypergraph Sidorenko's conjecture

Positive graphs have vanishing graph code density

Abstract

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it is symmetric, in the sense that it is formed by gluing two copies of some subgraph along an independent set. We prove several results relating to this conjecture. First, we make progress towards the conjecture itself by showing that any connected positive graph must have a vertex of even degree. We then make use of this result to identify some new counterexamples to the analogue of Sidorenko's conjecture for hypergraphs. In particular, we show that, for $r$ odd, every $r$-uniform tight cycle is a counterexample, generalising a recent result of Conlon, Lee and Sidorenko that dealt with the case $r=3$. Finally, we relate the positive graph conjecture to the emerging study of graph codes by showing that any positive graph has vanishing graph code density, thereby improving a result of Alon who proved the same result for symmetric graphs. Our proofs make use of a variety of tools and techniques, including the properties of independence polynomials, hypergraph quasirandomness and discrete Fourier analysis.