Fractal property of the graph homomorphism order

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 a more involved elementary argument.

Keywords

• graph homomorphism
• homomorphism order
• fractal property
• density
• universality