Cycles in 5-connected triangulations

Abstract

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Hamiltonian cycles in 4-connected plane triangulations with few 4-separators

“…We say S saturates a 4-or 5-cycle C if S contains two vertices of C. A diamond-6-cycle is the graph depicted in Figure 1, where the white vertices are called crucial. We say S saturates a diamond-6-cycle D if S contains three crucial vertices of D. Recall that the Euler genus eg(Σ) of a surface Σ is defined to be 2 − χ(Σ), where χ(Σ) denotes the Euler characteristic of The following tool is due to Alahmadi et al [1], which helps finding homotopic curves from a sufficiently large family of curves.…”

“…The next three lemmas aim at finding a vertex set that saturates no 4-cycle, or 5-cycle, or diamond-6-cycle. The main idea of the proofs comes from [1].…”

“…The proof of following lemma is omitted as it can be readily deduced from the proof of [1,Lemma 10].…”

“…One of the key ingredients of the proof of exponential lower bound on the number of hamiltonian cycles in 5-connected triangulations given by Alahmadi et al [1] is to find many edge sets F ⊆ E(G) so that G − F is 4-connected. Their approach was refined by [12,11] for 4-connected planar triangulations.…”

“…Let G be a triangulation of any surface and A ⊆ V (G) be a 3-separator of G. It is shown in the proof of [1,Lemma 1] and its subsequent discussion that G[A] is a surface separating 3-cycle. Using this fact and a theorem of Thomas and Yu [17] that every 4-connected projective planar graph is hamiltonian, the proof of [11, Lemma 2.1] can be easily modified to show the following result.…”