Consistent random vertex-orderings of graphs

TL;DR

This paper investigates the uniqueness of vertex-ordering distributions in graphs with hereditary properties, showing that most such properties only admit uniform orderings, with some exceptions.

Contribution

It characterizes when random vertex-orderings are uniquely uniform for hereditary graph properties and provides examples of non-uniform orderings.

Findings

Most hereditary properties have only uniform random vertex-orderings.

Identifies specific cases where non-uniform orderings exist.

Provides theoretical framework for understanding graph vertex-orderings.

Abstract

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist.