Chords of 2-factors in planar cubic bridgeless graphs

TL;DR

This paper characterizes edges in 2-edge-connected planar cubic graphs related to 2-factors and perfect matchings, providing structural insights and conjectures about cycle properties, with proofs involving vertex colorings of plane triangulations.

Contribution

It offers a complete characterization of edges in 2-edge-connected planar cubic graphs concerning 2-factors and perfect matchings, introducing new structural properties and conjectures.

Findings

Every edge in a 2-edge-connected planar cubic graph is either in a 2-edge-cut or a cycle chord in a 2-factor.

In cyclically 4-edge-connected planar cubic graphs (except K_2^3 and K_4), each edge is in a perfect matching whose removal disconnects the graph.

For any two edges on a facial cycle, there exists a 2-factor containing both edges in the same cycle.

Abstract

We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically 4-edge-connected planar cubic graph, except $K_2^3$ and $K_4$, is contained in a perfect matching whose removal disconnects the graph. We obtain a complete characterization of 2-edge-connected planar cubic graphs that have an edge such that every 2-factor containing the edge is a Hamiltonian cycle, and also of those that have an edge such that the complement of every perfect matching containing the edge is a Hamiltonian cycle. Another immediate consequence of the main result is that for any two edges contained in a facial cycle of a 2-edge-connected planar cubic graph, there exists a 2-factor in the graph such that both edges are contained in the same cycle of the 2-factor. We conjecture that this property holds for any two edges in a 2-edge-connected planar cubic graph, and prove it for planar cubic bipartite graphs. The main result is proved in the dual form by showing that every plane triangulation admits a vertex 3-coloring such that no face is monochromatic and there is exactly one specified edge between a specified pair of color classes.