Fractal property of the graph homomorphism order

TL;DR

This paper demonstrates that the homomorphism order of finite undirected graphs exhibits a fractal structure, where each interval is either universal or a gap, revealing complex and rich properties of graph homomorphisms.

Contribution

It establishes the fractal property of the homomorphism order, showing all intervals are either universal or gaps, using novel proofs including the Sparse Incomparability Lemma.

Findings

Intervals are either universal or gaps in the homomorphism order.

The fractal property explains the complex structure of graph homomorphisms.

The property is proven using both the Sparse Incomparability Lemma and elementary arguments.

Abstract

We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by more involved elementary argument.