### Abstract

Results already exist for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0001 and urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0002 for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0003, and so we focus on the intermediate cases. We establish that urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0004 whp (with high probability) when n ≪ m = n1 + o(1), that g(n,m) = (1 + o(1))μ(λ)m whp for a given function μ(λ) when m∼λn for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0005, and that urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0006 whp when urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0007 for n2/3 ≪ s ≪ n.

We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added. Given any connected graph with bounded maximum degree, we find that the addition of ϵn edges will whp result in a graph with genus Ω(n), even when ϵ is an arbitrarily small constant! We thus call this the “fragile genus” property.

Original language | English |
---|---|

Number of pages | 25 |

Journal | Random structures & algorithms |

DOIs | |

Publication status | E-pub ahead of print - 2019 |

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### Cite this

*Random structures & algorithms*. https://doi.org/10.1002/rsa.20871

**The genus of the Erdős-Rényi random graph and the fragile genus property.** / Dowden, C.; Kang, M.; Krivelevich, M.

Research output: Contribution to journal › Article › Research › peer-review

*Random structures & algorithms*. https://doi.org/10.1002/rsa.20871

}

TY - JOUR

T1 - The genus of the Erdős-Rényi random graph and the fragile genus property

AU - Dowden, C.

AU - Kang, M.

AU - Krivelevich, M.

PY - 2019

Y1 - 2019

N2 - We investigate the genus g(n,m) of the Erdős‐Rényi random graph G(n,m), providing a thorough description of how this relates to the function m = m(n), and finding that there is different behavior depending on which “region” m falls into.Results already exist for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0001 and urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0002 for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0003, and so we focus on the intermediate cases. We establish that urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0004 whp (with high probability) when n ≪ m = n1 + o(1), that g(n,m) = (1 + o(1))μ(λ)m whp for a given function μ(λ) when m∼λn for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0005, and that urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0006 whp when urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0007 for n2/3 ≪ s ≪ n.We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added. Given any connected graph with bounded maximum degree, we find that the addition of ϵn edges will whp result in a graph with genus Ω(n), even when ϵ is an arbitrarily small constant! We thus call this the “fragile genus” property.

AB - We investigate the genus g(n,m) of the Erdős‐Rényi random graph G(n,m), providing a thorough description of how this relates to the function m = m(n), and finding that there is different behavior depending on which “region” m falls into.Results already exist for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0001 and urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0002 for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0003, and so we focus on the intermediate cases. We establish that urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0004 whp (with high probability) when n ≪ m = n1 + o(1), that g(n,m) = (1 + o(1))μ(λ)m whp for a given function μ(λ) when m∼λn for urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0005, and that urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0006 whp when urn:x-wiley:rsa:media:rsa20871:rsa20871-math-0007 for n2/3 ≪ s ≪ n.We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added. Given any connected graph with bounded maximum degree, we find that the addition of ϵn edges will whp result in a graph with genus Ω(n), even when ϵ is an arbitrarily small constant! We thus call this the “fragile genus” property.

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85067414192&partnerID=MN8TOARS

U2 - 10.1002/rsa.20871

DO - 10.1002/rsa.20871

M3 - Article

JO - Random structures & algorithms

JF - Random structures & algorithms

SN - 1042-9832

ER -