On cyclically 4-connected cubic graphs
R. J. Kingan, S. R. Kingan · AKCE International Journal of Graphs and Combinatorics · 2024
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 Q 8 or Q 10 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.