On cyclically 4-connected cubic graphs

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 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.