Positive graphs

TL;DR

This paper investigates 'positive' graphs characterized by nonnegative homomorphism numbers into all edge-weighted graphs, proposing a conjecture about their structure, proving it for certain classes, and supporting it with computational verification up to 9 nodes.

Contribution

The paper introduces a conjecture on the structure of positive graphs, proves it for specific classes including all trees, and uses computational methods to verify it for small graphs.

Findings

Positive graphs have a homomorphic image with at least half the nodes.

All positive graphs up to 9 nodes satisfy the conjecture.

Positive graphs possess a homomorphic image with even pre-image counts for edges.

Abstract

We study "positive" graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary simple graph and gluing them together along an independent set of nodes. We prove the conjecture for various classes of graphs including all trees. We prove a number of properties of positive graphs, including the fact that they have a homomorphic image which has at least half the original number of nodes but in which every edge has an even number of pre-images. The results, combined with a computer program, imply that the conjecture is true for all graphs up to 9 nodes.