The Cycle Counts of Graphs
Combinatorics
2025-12-01 v3
Abstract
We prove that an inseparable graph can have any positive number of cycles with the six exceptions 2, 4, 5, 8, 9, 16, and that an inseparable cubic graph has the additional exceptions 1 and 13. The exceptions for simple inseparable cubic graphs are unknown.
Keywords
Cite
@article{arxiv.2507.02260,
title = {The Cycle Counts of Graphs},
author = {Ryan McCulloch and Brendan D. McKay and Alireza Salahshoori and Thomas Zaslavsky},
journal= {arXiv preprint arXiv:2507.02260},
year = {2025}
}
Comments
9 pp., 8 figures. v2: added data link. v3: 10 pp., improved writing; more computational details; added data for variant questions