Longer cycles in essentially 4-connected planar graphs

Abstract

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. Jackson and Wormald proved that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least 2n+4 5 , and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least 5 8 (n + 2). This improves the previously best-known lower bound 3 5 (n + 2).

Dynamics of Cycles in Polyhedra I: The Isolation Lemma

“…Regarding lower bounds, Jackson and Wormald [9] proved in 1992 that circ(G) ≥ 2 5 (n + 2) for every essentially 4-connected planar graph G on n vertices. Fabrici, Harant and Jendroľ [3] improved this lower bound to circ(G) ≥ 1 2 (n + 4); this result in turn was recently strengthened to circ(G) ≥ 3 5 (n + 2) [5], and then further to circ(G) ≥ 5 8 (n + 2) [6]. For the restricted case of maximal planar essentially 4-connected graphs, the matching lower bound circ(G) ≥ 2 3 (n + 4) was proven in [4]; however, the methods used there are specific to maximal planar graphs.…”

“…We do not require that these single vertices have degree three (this differs e.g. from [3,5,6]). A chord of a cycle C is an edge vw / ∈ E(C) for which v and w are in C. By a result of Whitney [13], every 3-connected planar graph has a unique embedding into the plane (up to flipping and the choice of the outer face).…”

“…Let a face f of H be thin if V − = ∅ and f ∈ F (H − ), and thick otherwise. A face f of H is called minor if it is either thin and incident to exactly one edge that is not in C or thick and incident to exactly one vertex of V − ∪ V + ; otherwise, f is called major (this definition differs from [3,5,6]). Our motivation for minor faces stems from the fact that all their C-vertices are consecutive in C, which we will later use to impose structural restrictions on the incident edges to these vertices.…”

“…While one part of the proof scheme for Lemma 1 follows the established approach of using Tutte cycles in combination with the discharging method, 1 personal communication with Jochen Harant we contribute an intricate intersection argument on the weight distribution between groups of neighboring faces, which we call tunnels. This method differs substantially from the ones used in [3,4,5,6] (for example, [4] exploits the inherent structure of maximal planar graphs) and is able to harness the dynamics of extending cycles in general polyhedral graphs. In particular, we will discharge weights along an unbounded number of faces, which was an obstacle that was not needed to overcome for the previous bounds.…”