Dvorak-Dell-Grohe-Rattan theorem via an asymptotic argument

TL;DR

This paper presents a new, simpler proof of a theorem linking graph isomorphism tests to homomorphism counts, using ordering WL-labels and asymptotic methods.

Contribution

It introduces an alternative proof of the Dvorak-Dell-Grohe-Rattan theorem based on asymptotic arguments, simplifying previous proofs.

Findings

New proof based on ordering WL-labels

Asymptotic argument simplifies the proof

Reinforces the connection between homomorphism counts and graph isomorphism

Abstract

Two graphs $G_1,G_2$ are distinguished by the Weisfeiler--Leman isomorphism test if and only if there is a tree $T$ that has a different number of homomorphisms to $G_1$ and to $G_2$. There are two known proofs of this fact -- a logical proof by Dvorak and a linear-algebraic proof by Dell, Grohe, and Rattan. We give another simple proof, based on ordering WL-labels and asymptotic arguments.