Chords of 2-Factors in Planar Cubic Bridgeless Graphs

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 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 with at least six vertices 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 in the case the graph is also bipartite. 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.