%A Dowden, Chris
%A Kang, Mihyun
%A Moßhammer, Michael
%A Sprüssel, Philipp
%D 2019
%T The evolution of random graphs on surfaces of non-constant genus
%K
%X 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 S g (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 S g (n,m) as m increases, focussing on the number |H 1 | of vertices in the largest component. For g=o(n) , we show that |H 1 | exhibits two phase transitions, one at around m=n/2 and a second one at around m=n . The exact behaviour of |H 1 | in the critical windows of these phase transitions depends on the order of g=g(n) .
%U http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1206
%J Acta Mathematica Universitatis Comenianae
%0 Journal Article
%P 631-636%V 88
%N 3
%@ 0862-9544
%8 2019-07-29