DOI: 10.5281/zenodo.17752422
title: Triangle-free CFS graphs (≤12 vertices) and associated RACG data

description: # 📁 Triangle-Free CFS Graphs ≤ 12 Vertices  ### with Right-Angled Coxeter Group (RACG) Invariants

This dataset contains **all triangle-free CFS graphs with up to 12 vertices**, together with a variety of geometric, combinatorial, and group-theoretic invariants of the **right-angled Coxeter groups** they define.  It supplements the computational and theoretical methods developed in the following preprint:

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## 📄 Reference Preprint

**RAAGedy right-angled Coxeter groups**  Christopher H. Cashen, Pallavi Dani, Alexandra Edletzberger, Annette Karrer  **arXiv:** 2506.16789  **DOI:** https://doi.org/10.48550/arXiv.2506.16789  **Subjects:** Group Theory (math.GR)  **MSC:** 20F65, 20F55  **Length:** 77 pages, 32 figures

The dataset complements the arguments in **Sections 4–6** of the preprint, particularly the analysis of:- RAAGedy components and obstructions,- coarse geometric tests for RACG-to-RAAG quasi-isometry,- computational verification for all graphs with ≤10 vertices (extended here to 12).

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## 📦 Contents

CFSdf_up_to_12.pkl ← pandas DataFrame (39 MB)CFS_database_management.py ← lookup and isomorphism utilitiesREADME.md ← this file

 

Each row in the `.pkl` represents **one isomorphism type** of a triangle-free CFS graph with ≤12 vertices.  All graphs are already filtered — no extra processing is required.

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## 🧪 Loading the Dataset

```pythonimport pickle

with open("CFSdf_up_to_12.pkl", "rb") as f:    data = pickle.load(f)

print(len(data), "graphs loaded.")print("Columns:", list(data.columns))

 

 

 

# Example: a square (C₄) — this *is* CFS and appears in the dataset.

import networkx as nximport CFS_database_management as cfs

G = nx.cycle_graph(4)  # 4-cycle C4

try:    idx = cfs.dfindex(data, G)    print("Found at index:", idx)    print(data.loc[idx])except IndexError:    print("Unexpected: C4 not found. (Should be present.)")

 

🧠 Selected DataFrame Columns| Column                                        | Purpose                                  || --------------------------------------------- | ---------------------------------------- || `graph`                                       | NetworkX graph object                    || `graph_hash`                                  | Degree-based coarse isomorphism hash     || `RAAGedy`, `Xi_RAAGedy_component`             | RAAGedy classification / component label || `Dani_Levcovitz`, `ZZ_obstruction`            | known obstructions from literature       || `ladder`, `near_double`, `strong`, `planar`   | structural conditions                    || `stable_cycle_depth`, `compliant_cycle_depth` | depth invariants                         || `iterated_splittings`, `goc`                  | splitting and geometric structure        || `nodes`, `edges`, `maxvalence`                | metadata                                 || `locally_connected_visual_boundary`           | RACG boundary property                   || `totally_disconnected_Morse_boundary`         | RACG boundary property                   |

 

🔧 Dependenciespip install pandas networkxpip install git+https://github.com/cashenchris/RACG.git

Everything needed for lookup and hashing is contained in the single fileCFS_database_management.py.

📌 Dataset Citation (placeholder)

When this dataset receives a Zenodo DOI, please cite:

Cashen, C. (2025).Triangle-free CFS graphs (≤12 vertices) and associated RACG data (v1.0).Dataset. TU Wien / Zenodo. DOI: TBA

📬 Contact

Christopher H. CashenTU WienInstitute of Discrete Mathematics and GeometryWiedner Hauptstraße 8–101040 Vienna, Austria

📧 christopher.cashen@tuwien.ac.at

🔗 https://github.com/cashenchris