On cyclically 4-connected cubic graphs

Abstract

AbstractA 3-connected cubic graph is cyclically 4-connected if it has at least [Formula: see text] 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 [Formula: see text] and [Formula: see text] denote the ladder and Möbius ladder on [Formula: see text] 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, [Formula: see text] or [Formula: see text] for [Formula: see text], or G is obtained from [Formula: see text] or [Formula: see text] by bridging pairs of edges with cycle spread at least [Formula: see text]. The concept of cycle spread also naturally leads to methods for constructing cyclically k-connected cubic graphs from smaller ones, but for [Formula: see text] the method is not exhaustive.

Keywords

• Cubic graphs
• cyclically 4-connected graphs
• snarks