Cubic Graphs and Related Triangulations on Orientable Surfaces

Fang, Wenjie; Kang, Mihyun; Moßhammer, Michael; Sprüssel, Philipp

doi:10.37236/5989

Cited by 8 publications

(4 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result is also claimed in [3] without a detailed proof, but the equation defining ρ m is incorrect, as well as the claimed value γ m ≈ 3.973.…”

Section: Theorem 13mentioning

confidence: 82%

Random cubic planar graphs revisited

Noy

Requilé

Rué

2016

Electronic Notes in Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…This result is also claimed in [3] without a detailed proof, but the equation defining ρ m is incorrect, as well as the claimed value γ m ≈ 3.973.…”

Section: Theorem 13mentioning

confidence: 82%

Random cubic planar graphs revisited

Noy

Requilé

Rué

2016

Electronic Notes in Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…Properties of random cubic planar graphs were studied in these works and also in [27,25]. The investigation of cubic planar graphs also stimulated further research directions, such as the study of 4-regular planar graphs [23] or cubic graphs on general orientable surfaces [13].…”

Section: Introductionmentioning

confidence: 98%

The Uniform Infinite Cubic Planar Graph

Stufler¹

2022

Preprint

View full text Add to dashboard Cite

We prove that the random simple connected cubic planar graph Cn with an even number n of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between Cn and a model where the networks inserted at the links of the largest 3-connected component of Cn are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest 3-connected component concentrates at κn for κ ≈ 0.85085, with Airy-type fluctuations of order n 2/3 . The second-largest component is shown to have significantly smaller size Op(n 2/3 ).

show abstract

“…In [23] the typical number of triangles in 3-connected cubic planar graphs was determined. Further research directions include 4-regular planar graphs [21], cubic graphs on general orientable surfaces [14], and cubic planar maps [12].…”

Section: Introductionmentioning

confidence: 99%

First-passage percolation on random simple triangulations

Stufler¹

2022

Preprint

View full text Add to dashboard Cite

We study first-passage percolation on random simple triangulations and their dual maps with independent identically distributed link weights. Our main result shows that the first-passage percolation distance concentrates in an op(n 1/4 ) window around a constant multiple of the graph distance.(1) The author's program simtria provides a highly optimized and efficient sampler for random simple triangulations. It supports multithreading and uses efficient memory management in order to facilitate large scale simulations. The program is open source and available for free on the github: https://github.com/BenediktStufler/simtria.

show abstract

Cubic Graphs and Related Triangulations on Orientable Surfaces

Cited by 8 publications

References 26 publications

Random cubic planar graphs revisited

Random cubic planar graphs revisited

The Uniform Infinite Cubic Planar Graph

First-passage percolation on random simple triangulations

Contact Info

Product

Resources

About