# Circumference of essentially 4-connected planar triangulations

@article{Fabrici2021CircumferenceOE, title={Circumference of essentially 4-connected planar triangulations}, author={Igor Fabrici and Jochen Harant and Samuel Mohr and Jens M. Schmidt}, journal={J. Graph Algorithms Appl.}, year={2021}, volume={25}, pages={121-132} }

A 3-connected graph G is essentially 4-connected if, for any 3-cut S ⊆ V (G) of G, at most one component of G− S contains at least two vertices. We prove that every essentially 4-connected maximal planar graph G on n vertices contains a cycle of length at least 2 3 (n+ 4); moreover, this bound is sharp.

## One Citation

Dynamics of Cycles in Polyhedra I: The Isolation Lemma

- MathematicsArXiv
- 2020

The first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes is taken, with a powerful and very general inductive motor for proving long cycles and computing them and improves the lower bound to match the upper.

## References

SHOWING 1-10 OF 29 REFERENCES

On Longest Cycles in Essentially 4-connected Planar Graphs

- MathematicsElectron. Notes Discret. Math.
- 2016

On the Circumference of Essentially 4-connected Planar Graphs

- MathematicsJ. Graph Algorithms Appl.
- 2020

It is proved that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least $\frac{5}{8}(n+2)$.

Nonhamiltonian 3-Connected Cubic Planar Graphs

- MathematicsSIAM J. Discret. Math.
- 2000

It is established that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian and this bound is determined to be sharp, and all nonhamiltonian examples of order 42 are presented.

On a Closure Concept in Claw-Free Graphs

- MathematicsJ. Comb. Theory, Ser. B
- 1997

A suucient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on ham Miltonicity in 2-tough graphs and the Hamiltonicity of 7-connected claw- free graphs are obtained as corollaries.

Longest paths and cycles in K1, 3-free graphs

- MathematicsJ. Graph Theory
- 1985

It is shown that the standard results concerning longest paths and cycles in graphs can be improved for K1,3-free graphs and conditions for the existence of a hamiltonian path and cycle are obtained.

Longest cycles in cyclically 4-edge-connected cubic planar graphs

- GeologyAustralas. J Comb.
- 2018

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