On cyclically 4-connected cubic graphs

Authors

R. J. Kingan, Bloomberg L.P.
S. R. Kingan, CUNY Brooklyn CollegeFollow

Document Type

Article

Publication Date

2024

Abstract

A 3-connected cubic graph is cyclically 4-connected if it has at least n ≥ 8 vertices and when removal of a set of three edges results in a disconnected graph, only one component has cycles. By introducing the notion of cycle spread to quantify the distance between pairs of edges, we get a new characterization of cyclically 4-connected graphs. Let Q_n and V_n denote the ladder and Möbius ladder on n ≥ 8 vertices, respectively. We prove that a 3-connected cubic graph G is cyclically 4-connected if and only if G is either the Petersen graph, Q_n or V_n for n ≥ 8, or G is obtained from Q₈ or Q₁₀ by bridging pairs of edges with cycle spread at least (1, 2). The concept of cycle spread also naturally leads to methods for constructing cyclically k-connected cubic graphs from smaller ones, but for k ≥ 5 the method is not exhaustive.

Comments

This article was originally published in AKCE International Journal of Graphs and Combinatorics, available at https://doi.org/10.1080/09728600.2024.2333397

This work is published under a Creative Commons Attribution-NonCommercial 4.0 International License.