Dynamics of Cycles in Polyhedra I: The Isolation Lemma
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A cycle C of a graph G is isolating if every component of G-V(C) is a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle C of length 8 ≤ |E(C)| < 2/3(|V(G)|+3) implies an isolating cycle C' of larger length that contains V(C). By “hopping” iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving and computing long cycles (we will give an algorithm with running time O(n^2)). This provides a method to prove lower bounds on Tutte cycles, as C' will be a Tutte cycle of G if C is. We also prove that E(C') ≤ E(C)+3 if G does not contain faces of size five, which gives a new tool for proving results about cycle spectra and evidence that these face sizes obstruct long cycles. As a sample application, we test our motor on a conjecture on essentially 4-connected graphs. A planar graph is essentially 4-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Essentially 4-connected graphs have been thoroughly investigated throughout literature as the subject of Hamiltonicity studies. Jackson and Wormald proved that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least 2/5(n+2), and this result has recently been improved multiple times, culminating in the lower bound 5/8(n+2). However, the best known upper bound is given by an infinite family of such graphs in which every graph G on n vertices has no cycle longer than 2/3(n+4); this upper bound is still unmatched. Using isolating cycles, we improve the lower bound to match the upper (up to a summand +1). This settles the long-standing open problem of determining the circumference of essentially 4-connected planar graphs.
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