Graph classes with given 3‐connected components: Asymptotic enumeration and random graphs

Abstract

We consider graph classes G in which every graph has components in a class C of connected graphs. We provide a framework for the asymptotic study of |G n,N |, the number of graphs in G with n vertices and N := λn components, where λ ∈ (0, 1). Assuming that the number of graphs with n vertices in C satisfies |C n | ∼ bn −(1+α) ρ −n n!, n → ∞ for some b, ρ > 0 and α > 1-a property commonly encountered in graph enumeration-we show that |G n,N | ∼ c(λ)n f (λ) (log n) g(λ) ρ −n h(λ) N n! N ! , n → ∞ for explicitly given c(λ), f (λ), g(λ) and h(λ). These functions are piecewise continuous with a discontinuity at a critical value λ * , which we also determine. The central idea in our approach is to sample objects of G randomly by so-called Boltzmann generators in order to translate enumer-ative problems to the analysis of iid random variables. By that we are able to exploit local limit theorems and large deviation results well-known from probability theory to prove our claims. The main results are formulated for generic com-binatorial classes satisfying the SET-construction.

The Maximum Degree of Random Planar Graphs

“…Theorem 5.4 from [9] immediately implies the following statement about the probability distribution of lb(C n ).…”

The maximum degree of random planar graphs

“…In the present paper, Boltzmann samplers also play a key role. A crucial fact, proved independently using probabilistic [24] and analytic methods [20], is that w.h.p. a connected random planar graph has a unique block (2-connected component) of linear size, and the remaining blocks are of order at most n 2/3 .…”

Further results on random cubic planar graphs

“…As we will see later, p can be computed exactly using the dissymmetry theorem. Once we have the value of p , a standard proof (see ) shows that the number of connected components in a random cubic graph is asymptotically distributed as X + 1, where X is a Poisson law of parameter λ ≈0.000604.…”

“…We have obtained similar results for parameters that have been studied for several classes of planar and related classes of graphs . We can show that the number of cut vertices, the number of isthmuses (separating edges), and the number of blocks (2‐connected components, including isthmuses) are all asymptotically normal with linear expectation and variance.…”