Chords of 2-factors in planar cubic bridgeless graphs [article]

Ajit Diwan
<span title="2021-10-25">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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
more &raquo; ... 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.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="">arXiv:2110.12584v1</a> <a target="_blank" rel="external noopener" href="">fatcat:4t4nnchg3zfqzm3gwx3dke36ai</a> </span>
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