Asymptotic enumeration and limit laws for graphs of fixed genus

Abstract

International audienceIt is shown that the number of labelled graphs with $n$ vertices that can be embedded in the orientable surface $\mathbb{S}_g$ of genus $g$ grows asymptotically like $ c^{(g)}n^{5(g-1)/2-1}\gamma^n n! $, where $c^{(g)} >0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case $g=0$, obtained by Giménez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in $\mathbb{S}_g$ has a unique 2-connected component of linear size with high probability

The evolution of random graphs on surfaces

“…, u h+1 (at most 7 3 n h possibilities, as in the previous proof, since we again need to look for what are now triangulated appearances of K 3 ). Thus, we find that we have built each graph at most 7 3 n h times. Hence, the number of distinct graphs (in S g (n, m)) that we have constructed must be at least |Gn|( αn 2 ) h+1…”

“…. , u a , we now have at most 7 3 n a possibilities instead of n a , since we need to look for what will be triangulated appearances of K 3 (note there are at most n 3 vertex-disjoint triangulated appearances of K 3 , and each can have a vertex in common with at most 6 others, since there are only 6 other triangles touching any triangulated appearance). Hence, we obtain…”

“…S 0 (n)) have now also been generalised to S g (n) [7,26,27]. However, similar extensions have not yet been achieved for the full S g (n, m) case, where the complexity of the general genus setting is combined with the extra restriction on the number of edges.…”

On the sum of digits of some sequences of integers

“…More generally, as it is shown in [2], see also [1], the number of labelled graphs which can be embedded in a surface of genus satisfies a very similar formula (with the same growth factor). See Table 1 for the asymptotics of these sequences.…”

“…Some of the above results were superseded by the results from the recent paper [8], where it is shown that if r = ( holds for almost all . Note that inequality (3) improves (1) for the case of the middle binomial coefficients B because C = S (r) for r = (2), as well as inequality (2) for the case of the Apéry numbers A because A = S (r) for r = (2 2).…”