Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 631-636
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Chris Dowden; Mihyun Kang; Michael Moßhammer; Philipp Sprüssel; Chris Dowden; Mihyun Kang; Michael Moßhammer; Philipp Sprüssel. The evolution of random graphs on surfaces of non-constant genus. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 631-636. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a42/
@article{AMUC_2019_88_3_a42,
author = {Chris Dowden and Mihyun Kang and Michael Mo{\ss}hammer and Philipp Spr\"ussel and Chris Dowden and Mihyun Kang and Michael Mo{\ss}hammer and Philipp Spr\"ussel},
title = { The evolution of random graphs on surfaces of non-constant genus},
journal = {Acta mathematica Universitatis Comenianae},
pages = {631--636},
year = {2019},
volume = {88},
number = {3},
url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a42/}
}
TY - JOUR
AU - Chris Dowden
AU - Mihyun Kang
AU - Michael Moßhammer
AU - Philipp Sprüssel
AU - Chris Dowden
AU - Mihyun Kang
AU - Michael Moßhammer
AU - Philipp Sprüssel
TI - The evolution of random graphs on surfaces of non-constant genus
JO - Acta mathematica Universitatis Comenianae
PY - 2019
SP - 631
EP - 636
VL - 88
IS - 3
UR - http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a42/
ID - AMUC_2019_88_3_a42
ER -
%0 Journal Article
%A Chris Dowden
%A Mihyun Kang
%A Michael Moßhammer
%A Philipp Sprüssel
%A Chris Dowden
%A Mihyun Kang
%A Michael Moßhammer
%A Philipp Sprüssel
%T The evolution of random graphs on surfaces of non-constant genus
%J Acta mathematica Universitatis Comenianae
%D 2019
%P 631-636
%V 88
%N 3
%U http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a42/
%F AMUC_2019_88_3_a42
Given a graph G, the genus of G denotes the smallest integer g for which G can be drawn on the orientable surface of genus g without crossing edges. For integers g,m≥0 and n>0, we let Sg(n,m) denote the graph taken uniformly at random from the set of all graphs on {1,2,...,n} with exactly m=m(n) edges and with genus at most g=g(n). We investigate the evolution of Sg(n,m) as m increases, focussing on the number |H1| of vertices in the largest component. For g=o(n), we show that |H1| exhibits two phase transitions, one at around m=n/2 and a second one at around m=n. The exact behaviour of |H1| in the critical windows of these phase transitions depends on the order of g=g(n).
