The Cycle Counts of Graphs

TL;DR

This paper characterizes the possible numbers of cycles in inseparable graphs, identifying specific exceptions for general and cubic cases, but leaves some cases for simple inseparable cubic graphs unresolved.

Contribution

It establishes the set of achievable cycle counts in inseparable graphs and cubic graphs, highlighting known exceptions and open problems.

Findings

Inseparable graphs can have any positive cycle count except 2, 4, 5, 8, 9, 16.

Inseparable cubic graphs have additional exceptions 1 and 13.

Exceptions for simple inseparable cubic graphs remain unknown.

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.