Document

# The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property

## File

LIPIcs.AofA.2018.17.pdf
• Filesize: 444 kB
• 13 pages

## Cite As

Chris Dowden, Mihyun Kang, and Michael Krivelevich. The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.17

## Abstract

We investigate the genus g(n,m) of the Erdös-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 behaviour depending on which `region' m falls into. Existing results are known for when m is at most n/(2) + O(n^{2/3}) and when m is at least omega (n^{1+1/(j)}) for j in N, and so we focus on intermediate cases. In particular, we show that g(n,m) = (1+o(1)) m/(2) whp (with high probability) when n << m = n^{1+o(1)}; that g(n,m) = (1+o(1)) mu (lambda) m whp for a given function mu (lambda) when m ~ lambda n for lambda > 1/2; and that g(n,m) = (1+o(1)) (8s^3)/(3n^2) whp when m = n/(2) + s for n^(2/3) << s << n. We then also show that the genus of fixed graphs 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 epsilon n edges will whp result in a graph with genus Omega (n), even when epsilon is an arbitrarily small constant! We thus call this the `fragile genus' property.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Random graphs
• Mathematics of computing → Graphs and surfaces
##### Keywords
• Random graphs
• Genus
• Fragile genus

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. A. Archdeacon and D. Grable. The genus of a random graph. Discrete Math., 142:21-37, 1995.
2. T. Bohman, A. Frieze, and R. Martin. How many random edges make a dense graph hamiltonian? Random Struct. Algor., 22:33-42, 2003.
3. B. Bollobás. Random Graphs. Cambridge University Press, Cambridge, 2001.
4. C. Dowden, M. Kang, and P. Sprüssel. The evolution of random graphs on surfaces. SIAM J. Discrete Math., 32:695-727, 2018.
5. A. Frieze and M. Karoński. Introduction to Random Graphs. Cambridge University Press, Cambridge, 2015.
6. S. Janson, T. Łuczak, and A. Ruciński. Random Graphs. Wiley, New York, 2000.
7. M. Kang, M. Moßhammer, and P. Sprüssel. Phase transitions in graphs on orientable surfaces. submitted, arXiv:1708.07671.
8. M. Krivelevich, M. Kwan, and B. Sudakov. Cycles and matchings in randomly perturbed digraphs and hypergraphs. Combin. Probab. Comput., 25:909-927, 2016.
9. M. Krivelevich, M. Kwan, and B. Sudakov. Bounded-degree spanning trees in randomly perturbed graphs. SIAM J. Discrete Math., 31:155-171, 2017.
10. M. Krivelevich and A. Nachmias. Colouring complete bipartite graphs from random lists. Random Struct. Algor., 29:436-449, 2006.
11. M. Krivelevich, D. Reichman, and W. Samotij. Smoothed analysis on connected graphs. SIAM J. Discrete Math., 29:1654-1669, 2015.
12. T. Łuczak. Component behaviour near the critical point of the random graph process. Random Struct. Algor., 1:287-310, 1990.
13. T. Łuczak. Cycles in a random graph near the critical point. Random Struct. Algor., 2:421-439, 1991.
14. T. Łuczak, B. Pittel, and J. C. Wierman. The structure of a random graph near the point of the phase transition. Trans. Amer. Math. Soc., 341:721-748, 1994.
15. M. Noy, V. Ravelomanana, and J. Rué. The probability of planarity of a random graph near the critical point. Proc. Amer. Math. Soc., 143:925-936, 2015.
16. V. Rödl and R. Thomas. On the genus of a random graph. Random Struct. Algor., 6:1-12, 1995.