[Paper Review] Cycle lengths modulo $k$ in large 3-connected cubic graphs

Kasper Szabo Lyngsie, Martin Merker|arXiv (Cornell University)|Apr 10, 2019

Advanced Graph Theory Research20 references4 citations

TL;DR

This paper proves that for any odd integer $k$ and natural number $m$, every sufficiently large 3-connected cubic graph contains a cycle whose length is congruent to $m$ modulo $k$. The result is sharp for 2-connected cubic graphs, as a constructed family demonstrates the failure of the property when $m$ and $k$ are both divisible by 3 and $k \geq 12$. The key contribution is establishing a threshold $N(k)$ beyond which the cycle length condition is guaranteed in 3-connected cubic graphs.

ABSTRACT

We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a family of graphs showing that this is not true for 2-connected cubic graphs if $m$ and $k$ are divisible by 3 and $k\geq 12$.

Motivation & Objective

• To determine under what conditions cycle lengths modulo $k$ exist in large 3-connected cubic graphs.
• To establish a threshold $N(k)$ such that all 3-connected cubic graphs with at least $N(k)$ vertices contain a cycle of length $m \mod k$ for any $m$ and odd $k$.
• To investigate whether the result extends to 2-connected cubic graphs, particularly when $m$ and $k$ are divisible by 3.
• To construct a family of 2-connected cubic graphs that violate the cycle length modulo $k$ condition under specific divisibility constraints.

Proposed method

• Use of structural graph theory to analyze cycle space and connectivity properties in cubic graphs.
• Application of known results on cycle covers and cycle space bases in 3-connected cubic graphs.
• Construction of a specific family of 2-connected cubic graphs to serve as counterexamples when $m$ and $k$ are divisible by 3 and $k \geq 12$.
• Employment of bounds on graph diameter and vertex count to establish the existence of cycles modulo $k$.
• Leveraging Bernoulli’s inequality in asymptotic estimates to bound the required number of vertices $N(k)$.
• Use of extremal graph theory techniques to show that large 3-connected cubic graphs must contain cycles of prescribed lengths modulo $k$.

Experimental results

Research questions

• RQ1 Does every sufficiently large 3-connected cubic graph contain a cycle of length congruent to $m$ modulo $k$ for any fixed $m$ and odd $k$?
• RQ2 What is the minimal number $N(k)$ such that all 3-connected cubic graphs with at least $N(k)$ vertices contain a cycle of length $m \mod k$?
• RQ3 Can the result be extended to 2-connected cubic graphs under the same conditions?
• RQ4 Under what conditions on $m$ and $k$ does the cycle length modulo $k$ property fail in 2-connected cubic graphs?
• RQ5 Is there a structural obstruction to the existence of such cycles when $m$ and $k$ are both divisible by 3 and $k \geq 12$?

Key findings

• For any odd $k$ and natural number $m$, there exists a threshold $N(k)$ such that all 3-connected cubic graphs with at least $N(k)$ vertices contain a cycle of length congruent to $m$ modulo $k$.
• The paper constructs a family of 2-connected cubic graphs that do not contain cycles of length $m \mod k$ when $m$ and $k$ are divisible by 3 and $k \geq 12$, showing the result does not extend to 2-connected graphs.
• The diameter of any such 3-connected cubic graph $G$ is at least $f(k)$, indicating a lower bound on structural spread.
• The bound $N(k)$ is implicitly defined, with the estimate $162^{13} < 10^{29}$ suggesting a very large but finite threshold for large $k$.
• Bernoulli’s inequality is used to derive asymptotic bounds on $N(k)$, supporting the existence of the required cycles in large graphs.
• The result establishes a sharp dichotomy: the property holds for 3-connected graphs but fails for 2-connected graphs under specific divisibility conditions.