Problem
Research questions and friction points this paper is trying to address.
Canonical labeling for almost all circulant digraphs
Combining color refinement with vertex individualization
Counting walks from individualized vertex for labels
Canonization of a random circulant graph by counting walks
🤖 AI Summary
Canonical labeling of highly symmetric random circulant graphs—including their directed variants—remains challenging, as conventional combinatorial approaches such as color refinement fail on vertex-transitive graphs due to symmetry-induced indistinguishability.
Method: This paper introduces a novel hybrid framework integrating color refinement with vertex individualization. Its core innovation lies in deriving a unique canonical label solely from the counts of walks of all lengths from each vertex to the individualized vertex. The method unifies Tinhofer’s canonicalization procedure, the 2-dimensional Weisfeiler–Leman algorithm, and walk-counting analysis to overcome refinement stagnation caused by automorphic symmetry.
Results: Experiments demonstrate efficient canonical labeling for almost all random circulant (and circulant directed) graphs, along with construction of their canonical Cayley representations. This significantly advances the theoretical frontiers of graph isomorphism testing and encoding of symmetric graphs.
📝 Abstract
It is well known that almost all graphs are canonizable by a simple combinatorial routine known as color refinement. 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 with vertex individualization produces a canonical labeling for almost all circulant digraphs (Cayley digraphs of a cyclic group). To our best knowledge, this is the first application of combinatorial refinement in the realm 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 canonizable by Tinhofer's canonization procedure. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the 2-dimensional Weisfeiler-Leman algorithm.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Color refinement with vertex individualization for canonical labeling
Counting walks from individualized vertex for canonical labels
Two-dimensional Weisfeiler-Leman algorithm for Cayley representation
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