Dynamics of Cycles in Polyhedra I: The Isolation Lemma

A cycle $C$ of a graph $G$ is \emph{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 $6 \leq |E(C)| < \left \lfloor \frac{2}{3}(|V(G)|+4) \right \rfloor$ 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 long cycles and computing them (we will give an algorithm with quadratic running time). This is the first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes. Our motor provides also a method to prove linear lower bounds on the length of Tutte cycles, as $C'$ will be a Tutte cycle of $G$ if $C$ is. We prove in addition that $|E(C')| \leq |E(C)|+3$ if $G$ contains no face of size five, which gives a new tool for results about cycle spectra, and provides evidence that faces of size five may obstruct long cycles in many graph classes. We test our motor on the following conjecture about essentially 4-connected graphs. A planar graph is \emph{essentially $4$-connected} if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{5}(n+2)$, and this result has recently been improved multiple times, culminating in the lower bound $\frac{5}{8}(n+2)$. However, the currently best known upper bound is given by an infinite family of such graphs in which no graph $G$ contains a cycle that is longer than $\left \lfloor \frac{2}{3}(n+4) \right \rfloor$; this upper bound is still unmatched. Using isolating cycles, we improve the lower bound to match the upper. All our results are tight.

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