Some remarks on the uncolored versions of the original CFI-graphs

TL;DR

This paper demonstrates that uncolored versions of the original CFI-graphs retain key properties and can be characterized by first-order formulas, simplifying their use in graph isomorphism and logic applications.

Contribution

It shows that uncolored CFI-graphs preserve essential features of the colored versions and can be described by first-order formulas, avoiding size blow-ups.

Findings

Uncolored CFI-graphs maintain the core properties of their colored counterparts.

A first-order formula can identify vertices of the same color in uncolored CFI-graphs.

Uncolored variants are effective substitutes in applications requiring color information.

Abstract

The CFI-graphs, named after Cai, F\"urer, and Immerman, are central to the study of the graph isomorphism testing and of first-order logic with counting. They are colored graphs, and the coloring plays a role in many of their applications. As usual, it is not hard to remove the coloring by some extra graph gadgets, but at the cost of blowing up the size of the graphs and changing some parameters of them as well. This might lead to suboptimal combinatorial bounds important to their applications. Since then for some uncolored variants of the CFI-graphs it has been shown that they serve the same purposes. We show that this already applies to the graphs obtained from the original CFI-graphs by forgetting the colors. Moreover, we will see that there is a first-order formula $\varphi(x,y)$ expressing in almost all uncolored CFI-graphs that $x$ and $y$ have the same color in the corresponding colored graphs.