Canonization of a random circulant graph by counting walks

TL;DR

This paper demonstrates that combining color refinement with vertex individualization effectively canonizes almost all circulant digraphs, revealing insights into their symmetry and automorphism structures, and introduces walk-counting as a key technique.

Contribution

It proves that a simple combinatorial approach suffices for canonical labeling of most circulant graphs, providing new evidence of the method's effectiveness on highly symmetric graphs.

Findings

Color refinement plus individualization canonizes almost all circulant digraphs.

Walk counting from a vertex to an individualized vertex determines its canonical label.

Almost all circulant graphs are shown to be Tinhofer-compact.

Abstract

It is well known that almost all graphs are canonizable by a simple combinatorial routine known as color refinement, also referred to as the 1-dimensional Weisfeiler-Leman algorithm. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of color refinement and vertex individualization yields a canonical labeling for almost all circulant digraphs (i.e., Cayley digraphs of a cyclic group). This result provides first evidence of good average-case performance of combinatorial refinement within the class of vertex-transitive graphs. Remarkably, we do not even need the full power of the color refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are compact in the sense of Tinhofer, that is, their polytops of fractional automorphisms are integral. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the more powerful 2-dimensional Weisfeiler-Leman algorithm.