The Cycle Counts of Graphs

TL;DR

The paper determines which positive integers occur as the number of cycles in inseparable graphs, proving that only 2,4,5,8,9,16 are impossible (with 1 and 13 also excluded for inseparable cubic graphs). It develops a two-pronged approach: a constructive, ear-based analysis for small counts and a general tree-subtree correspondence via outerplanar, inseparable graphs to realize large counts, supplemented by computer verification for a finite set up to 89. The results extend to inseparable Hamiltonian planar graphs and yield nonplanarity-related consequences, along with a suite of conjectures about other graph classes. The work combines classical graph-theoretic tools with computational verification to map cycle-count possibilities across several graph families.

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.

Figures (8)

• Figure 1: The two-ear graphs obtained by adding an ear to a theta graph. An asterisk $*$ denotes a path that may have length 0.
• Figure 2: Graphs that prove 10, …, 15 and 17, 18, 19 are cycle count numbers.
• Figure 3: The 1-ear extensions of $\widehat{K}_4$ and their cycle counts.
• Figure 4: The graphs show all ways to add a third ear to $\varTheta_4'$ in Figure \ref{['2-ears']} without creating a $\widehat{K}_4$ subgraph. The labels in (a2) show how it is the same as (a1). An asterisk $*$ denotes a path that may have length 0.
• Figure 5: Ear additions with four ears. The graphs show all ways to add a fourth ear to a graph in Figure \ref{['3-ears']} without creating a $\widehat{K}_4$ subgraph. An asterisk $*$ denotes a path that may have length 0.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3: Whitney
• Lemma 4: Ear-Path Lemma
• Proposition 5: Isotonicity
• proof
• Lemma 6: Extremality Lemma
• proof
• Lemma 7
• proof