Abstract
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 (Formula presented.) and (Formula presented.) for (Formula presented.), and so we focus on the intermediate cases. We establish that (Formula presented.) 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 (Formula presented.), and that (Formula presented.) whp when (Formula presented.) 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 |
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Pages (from-to) | 97-121 |
Number of pages | 25 |
Journal | Random Structures and Algorithms |
Volume | 56 |
Issue number | 1 |
DOIs | |
State | Published - 1 Jan 2020 |
Keywords
- fragile genus
- genus
- random graphs
All Science Journal Classification (ASJC) codes
- Software
- Applied Mathematics
- General Mathematics
- Computer Graphics and Computer-Aided Design