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

Lo, On-Hei Solomon

doi:10.1016/j.disc.2020.112126

Cited by 7 publications

(12 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following result for 5-connected planar triangulations was proved by Alahmadi et al [1]. Lo [6] observed that it is essentially true for certain 4-connected planar triangulations. We further observe that a slight variation holds for 4-connected planar triangulations with minimum degree 5.…”

Section: Preliminariesmentioning

confidence: 82%

“…Lo [6] also observed that the following lemma stated for 5-connected planar triangulations in [1] actually holds for 4-connected planar triangulations. Lemma 2.3 (Alahmadi, Aldred, and Thomassen, 2020; Lo, 2020).…”

Section: Preliminariesmentioning

confidence: 89%

“…Lemma 2.2 (Lo, 2020). Let G be a 4-connected planar triangulation and let S be an independent set of vertices of degree at most 6 in G, such that S saturates no 4-cycle in G. Then there exists a subset S ⊆ S of size at least |S|/541 such that S saturates no 5-cycle in G.…”

Section: Preliminariesmentioning

confidence: 99%

“…Lo [6] proved a lemma implying that any 4-connected planar triangulation has a large independent set or contains two vertices with a lot of common neighbors. Lemma 2.4 (Lo, 2020). Let G be a 4-connected planar triangulation.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recently, Lo [6] showed that every 4-connected n-vertex planar triangulation with O(log n) separating 4-cycles has Ω((n/ log n) 2 ) Hamiltonian cycles. In this paper, we improve Lo's result by weakening its hypothesis and strengthening its conclusion to a quadratic bound.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Number of Hamiltonian cycles in planar triangulations

Liu¹,

Yu²

2021

Preprint

View full text Add to dashboard Cite

Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if G is a 4-connected planar triangulation with n vertices then G contains at least 2(n−2)(n−4) Hamiltonian cycles, with equality if and only if G is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar n-vertex triangulations G that do not have too many separating 4-cycles or have minimum degree 5. We show that if G has O(n/log 2 n) separating 4-cycles then G has Ω(n 2 ) Hamiltonian cycles, and if δ(G) ≥ 5 then G has 2 Ω(n 1/4 ) Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a "double wheel" structure, providing further evidence to the above conjecture.

show abstract

Section: Preliminariesmentioning

confidence: 82%

Section: Preliminariesmentioning

confidence: 89%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations