Longest and shortest cycles in random planar graphs

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in $P(n,m)$ in the critical range when $m=n/2+o(n)$. In addition, we describe the block structure of $P(n,m)$ in the weakly supercritical regime when $n^{2/3}\ll m-n/2\ll n$.

Perhaps the most interesting case in Theorem 1.2 is when s 3 n −2 → ∞ (in the so-called weakly supercritical regime): whp the longest cycle is contained in the largest component L 1 . Moreover, the length of the shortest cycle in L 1 is of the same asymptotic order as the length of the longest cycle outside L 1 . In other words, there exists a 'threshold' function f (n) := ns −1 in the sense that whp for all cycles K in G we have where |K | denotes the length of K . An important structure related to cycles are blocks, because every cycle is contained in a block: a block of a graph H is a maximal 2-connected subgraph of H . We emphasise that we do not consider a bridge, i.e. an edge whose deletion increases the number of components, to form a block. Łuczak [28] investigated the block structure of G(n, m) when m = n/2+s for s 3   We note that Theorems 1.2(c) and 1.3 imply that whp the longest cycle lies in the largest block B 1 (G) and its length grows asymptotically like b 1 (G).
In the last few decades various models of random graphs have been introduced by imposing additional constraints to G(n, m), e.g. topological constraints or degree restrictions. Particularly interesting models are random planar graphs and, more generally, random graphs on surfaces which have attained considerable attention [8][9][10]15,21,23,32], since the pioneering work of McDiarmid, Steger, and Welsh [33] on random planar graphs and that of McDiarmid [31] on random graphs on surfaces. Many exciting results have been obtained, revealing richer and more complex behaviour. In particular, results are often found to feature thresholds, meaning that the probabilities of various properties change dramatically according to which 'region' the edge density falls into.
A natural question is whether random planar graphs satisfy similar properties as in Theorems 1.2 and 1.3. Kang and Łuczak [21] showed that the component structure of a random planar graph P (n, m) changes drastically when m ∼ n/2, analogously to G(n, m). In contrast, not much is known about the cycle and block structure of P (n, m). In this paper we investigate this open problem, determining the length of the shortest and longest cycle in P (n, m) and the order of blocks in P (n, m), in the light of Theorems 1.2 and 1.3.

Main results.
Throughout this section, we let P (n, m) denote the class of all vertex-labelled planar graphs on vertex set [n] with m = m(n) edges and P (n, m) be a graph chosen uniformly at random from P (n, m), denoted by P (n, m) ∈ R P (n, m).
As we will see in Theorem 4.5, Theorem 1.4 holds for a more general universal class of graphs, the so-called kernel-stable classes of graphs (see Definition 4.1 for a formal definition), which include the class of series-parallel graphs, the class of planar graphs, and the class of graphs on a surface, to mention a few.
Our second main result deals with the block structure of P (n, m). Due to Theorem 1.4(a) and (b) we focus on the weakly supercritical regime and will show that whp P (n, m) contains a unique largest block B 1 which is significantly larger than all other blocks, similarly as in G(n, m). However, the largest component in P (n, m) contains 'many' blocks, while the largest component in G(n, m) contains only a bounded number of blocks (cf. Theorem 1.3(c)). It is well known that when s 3 n −2 → −∞ or s = O n 2/3 , the probability that G(n, m) is planar is bounded away from 0 (see e.g. Theorem 4.9(a) and [18,29,36]). Hence, each graph property that holds whp in G(n, m) is also true whp in P (n, m). In particular, this implies that the cycle structure of P (n, m) 'behaves' similarly like that of G(n, m) (see Theorems 1.2(a), (b) and 1.4(a), (b)). However, when s 3 n −2 → ∞, whp G(n, m) is not planar (see e.g. [29,36]) and therefore, G(n, m) and P (n, m) can exhibit different asymptotic behaviours. Theorems 1.4(c) and 1.5 indicate that in view of the cycle and block structure this is indeed the case. For example, a 'threshold' function in the sense of (1) does not exist in P (n, m), because g (L 1 ) = Θ p ns −1 ≪ c (R) = Θ p n 1/3 . However, whp the longest cycle in P (n, m) is still contained in the largest component.
Kang and Łuczak [21] proved that in the case of s 3 n −2 → ∞ the core, i.e. the maximal subgraph of minimum degree at least two (also known as 2-core), is much smaller in P (n, m) compared to G(n, m). More precisely, whp the core of P (n, m) is of order Θ sn −1/3 , while that of G(n, m) is of order Θ s 2 n −1 . This has a natural impact on the order of the longest cycle and largest block in P (n, m) (cf. Theorems 1.2(c), 1.4(c) and Theorems 1.3(a) and 1.5(a)): c (P (n, m)) = O sn −1/3 whp ≪ c (G(n, m)) = Θ s 2 n −1 whp b 1 (P (n, m)) = Θ p sn −1/3 ≪ b 1 (G(n, m)) = Θ s 2 n −1 whp.
Furthermore, it is known that the 'edge density' in the part without the largest component is typically much larger in P (n, m) than in G(n, m) (see e.g. [23,Theorem 1.7]). This affects the order of the longest cycle outside the largest component (cf. Theorems 1.2(c), 1.4(c)): c (R(P (n, m))) = Θ p n 1/3 ≫ c (R(G(n, m))) = Θ p ns −1 .

Related work.
The so-called n-vertex model of a random planar graph is a graph chosen uniformly at random from the class of all vertex-labelled planar graphs on vertex set [n], denoted by P (n) ∈ R P (n). Giménez and Noy [15] showed that whp P (n) has (1 + o(1)) κn edges for a constant κ ≈ 2.21, i.e. P (n) 'behaves' like P (n, m) where m ≈ κn. Many exciting results on the block structure of P (n) were obtained in recent literature, revealing a different behaviour from that of P (n, m) as observed in Theorem 1.5. For example, Panagiotou and Steger [38] proved that whp b 1 (P (n)) = Θ(n). Later, Giménez, Noy, and Rué [16,Proposition 5.3] established that in fact an Airy-type central limit theorem holds for b 1 (P (n)). Stufler [41,Theorem 6.20, Corollary 6.42] determined the limiting distribution of b i (P (n)) for any fixed integer i ≥ 2, showing, among others, that b i (P (n)) = O p n 2/3 . Furthermore, Stufler [40,Remark 9.13] provided a detailed structural description of the graph P (n) \ B 1 (P (n)), i.e. P (n) without its largest block. Another model related to P (n, m) is the random connected planar graph C (n, m), which is a graph chosen uniformly at random from the class of all connected planar graphs on vertex set [n] having m = m(n) edges. Panagiotou [37,Theorem 1,Corollary 1] proved that if m = ⌊cn⌋ for a constant c ∈ (1, 3), then C (n, m) has a block of linear order. Moreover, there is a discussion in [16, End of Section 5] sketching how much stronger results on the order of the largest block in C (n, m) can in principle be obtained.

Key techniques.
One of the main proof techniques is the so-called core-kernel approach. We decompose a graph into the simple part (in which each component contains at most one cycle) and the complex part (in which each component contains at least two cycles). Then we decompose the complex part into its core and then into its kernel, a key structure obtained from the core by replacing each path whose internal vertices all have degree exactly two by an edge. Conversely, each graph can be uniquely constructed from the kernel by first subdividing the edges of the kernel, thereby obtaining the core, then replacing vertices of the core with rooted trees and adding the simple part (see Section 2.3).
In order to investigate the cycle and block structure of a random planar graph P = P (n, m), we begin with the analysis of the structure of its core C (P ), which is itself a random graph. Instead of directly analysing the random core C (P ), we introduce an auxiliary random core modelC , in which we split the 'randomness' into smaller parts. More precisely, we choose randomly a kernel and then randomly a subdivision number which is a total number of vertices that will be used for a subdivision of the kernel. Given these two random bits (i.e. a random kernel and a random subdivision number) we then randomly construct a core by randomly inserting vertices on the edges of the kernel. A crucial technique to analyse the random coreC is the famous Pólya urn model: We derive results on the maximum and minimum number of drawn balls of some colour in order to determine the length of the longest and shortest cycles in the core, respectively.
1.5. Outline of the paper. The rest of the paper is organised as follows. After providing the necessary notations, definitions, and concepts in Section 2, we present our proof strategy in Section 3. In Section 4 we define kernel-stable classes of graphs. In Section 5 we provide results on the Pólya urn model, which we use in Section 6 to derive the cycle structure of a core randomly built from a fixed kernel and a fixed subdivision number. Section 7 is devoted to a random kernel and Section 8 to the block structure of a random planar graph. In Sections 9 and 10 we provide the proofs of our main and auxiliary results, respectively. Finally in Section 11, we discuss various questions that remain unanswered. Next, we introduce some notion for random graphs which have the 'same' asymptotic behaviour in the sense that they are indistinguishable in view of properties that hold whp. Definition 2.5. For each n ∈ N, let G n and H n be random graphs. We say that G n and H n are contiguous if for every graph property Q 2.2. Weighted Multigraphs. Throughout the paper, we always assume implicitly that multigraphs are weighted by the so-called compensation factor, which was first introduced by Janson, Knuth, Łuczak, and Pittel [18].
For a finite class A of multigraphs we define To obtain the kernel K (H ), we replace any such path in the core C (H ) by an edge v 0 v i . By doing that, loops and multiple edges can be created and therefore in general the kernel K (H ) is a multigraph. Finally, we reverse the above decomposition and note that we can construct the core C (H ) by subdividing the edges of the kernel K (H ) with additional vertices, thereby ensuring that no loops and multiple edges of the kernel survive in the core. The number of additional vertices that are used to subdivide the kernel K (H ) to obtain the core C (H ) is called the subdivision number, denoted by S (H ), i.e.
2.4. Asymptotic notation. We will study asymptotic properties of random graphs on vertex set [n] as n tends to ∞, and all asymptotics are taken with respect to n. In addition to the standard Landau notation, we will use the notations by Janson [17] to express asymptotic orders of random variables.
Definition 2.7. Let (X n ) n∈N be a sequence of random variables and f : N → R ≥0 . Then, we write Moreover, for a sequence (A n ) n∈N of events we say that the probability that A n occurs is bounded away from 0 and 1 if lim inf n→∞ P [A n ] > 0 and lim sup n→∞ P [A n ] < 1, respectively.
We note that X n = O p f if and only if P |X n | ≥ h(n) f (n) = o(1) for any function h = ω (1). Similarly, we have X n = Ω p f if and only if P |X n | ≤ f (n)/h(n) = o(1) for any function h = ω (1). Furthermore, the statement X n = O p f is slightly weaker than having whp X n = O f . The latter says that there exists a constant c > 0 such that for every δ > 0 there is a N ∈ N satisfying P |X n | ≤ c f (n) ≥ 1−δ for all n ≥ N . In other words, we have a uniform constant c if whp X n = O f , while c may depend on δ in the case of X n = O p f .

PROOF STRATEGY
In order to analyse the cycle and block structure of a random planar graph P = P (n, m), we decompose P into smaller parts: We first decompose P into the complex part Q (P ), the core C (P ), and the kernel K (P ) as in Section 2.3. Moreover, we obtain the core C (P ) by subdividing the edges of the kernel K (P ) by S (P ) many additional vertices. We note that all blocks and cycles that do not lie in unicyclic components are contained in the core C (P ). Therefore, it is crucial to understand the cycle and block structure of the core C (P ). To this end, we introduce an auxiliary random core C which is created stepwise as follows. We choose randomly a typical kernel K and a typical subdivision number k. Then we construct C from K by randomly subdividing the edges of K with k additional vertices. In order to analyse the cycle and block structure of C (P ), we ask the following questions. (a) Which properties do a typical kernel K and a typical subdivision number k have, in particular with respect to cycles and blocks? (b) How do these properties translate to C when choosing a random subdivision? (c) How can we relate the random graph C to the original core C (P ) where we first choose the random graph P and then extract (deterministically) the core C (P ). In the rest of this section we will give an overview how to deal with these questions. We will start by considering question (b) in Section 3.1. The main idea is to find a relationship between a random core C and the Pólya urn model. Next, we will deal with question (a). We note that already lots of results about a typical kernel K and a typical subdivision number k are known, e.g. asymptotic order of v (K ) , e (K ), and k (see Theorem 4.8). In contrast, there are no results on the cycle and block structure of K known. We will deal with these open problems as follows. Firstly, we will show that a typical kernel 'behaves' asymptotically like a random cubic (i.e. 3-regular) planar multigraph (see Lemma 7.1). Then we will analyse the cycle (in Section 3.2) and block structure (in Section 3.3) of a random cubic planar multigraph by double counting arguments. Finally, we will introduce the concept of conditional random graphs in Section 3.4 to give an answer to question (c).
3.1. Random core and Pólya urn model. Given a (typical) kernel K and a (typical) subdivision number k ∈ N we let C = C (K , k) be a random core chosen uniformly at random from the set of all cores with kernel K and subdivision number k. In order to study the cycle and block structure of a random core C , we consider an auxiliary random multigraphC that 'behaves' similarly like C and is easier to study. We randomly place k additional labelled vertices {v 1 , . . . , v k } one after another on the edges e 1 , . . . , e N of K to obtainC . More precisely, in the i -th step of the random process we choose uniformly at random an edge of the current multigraph and place the vertex v i on that edge. We note that loops and multiple edges are allowed inC , although they do not appear in C . However, we will show that C andC behave asymptotically quite similarly (see Lemma 6.4 and Corollary 6.5). Thus, it suffices to consider the cycle and block structure ofC instead of C .
We note that each cycle ofC is a cycle in the kernel K together with some additional vertices placed on the edges. Thus, we are interested in the distribution of (X 1 , . . . , X N ), where X i is the number of vertices placed on edge e i . We observe that we can model the random vector (X 1 , . . . , X N ) with the following Pólya urn model (see e.g. [20,30] for details of Pólya urns). We start with N balls of N distinct colours F 1 , . . . , F N in a urn, where colour F i represents edge e i . Then we draw one ball uniformly at Use of a Pólya urn to construct a random core by sequentially subdividing the edges of a kernel with N = 5 edges by k = 10 additional vertices. The left-hand side represents the situation at the beginning, the right-hand side after four drawings.
random from the urn, say F j , and subdivide the corresponding edge e j by vertex v 1 . By doing so the edge e j is split into two new edges. Hence, we need in the next step two balls of the colour corresponding to e j . Therefore, we return the drawn ball to the urn along with an additional ball of the same colour. We repeat that procedure k times and observe that the number of drawn balls (after k steps) of colour F i is distributed like X i for each i ∈ [N ] (see also Figure 1). The Pólya urn model provides bounds on min 1≤i ≤ f X i and max 1≤i ≤ f X i for various f satisfying 1 ≤ f ≤ N (see Theorem 5.1). Assuming e 1 , . . . , e λ are the loops of K , we will derive bounds on the length of the shortest and longest cycle inC (see Lemma 6.7), denoted by g C and c C , by applying the following inequalities: Next, we consider how the block structure of K translates to the block structure ofC . We observe that each block inC is a block or loop in K together with additional vertices placed on the edges. So in general we can use similar ideas as for the cycle structure above. However, we need to slightly modify our Pólya urn model (see Section 5.2). For the cycle structure, we use many 'small' cycles (in fact, loops) simultaneously, but for the block structure we will fix one 'large' block B of K and consider how many vertices are placed on the edges of B . Therefore, we just need balls of two different colours, one representing edges in B and the other representing edges outside B . Using standard results on this Pólya urn model, we will show that each 'large' block B of K translates to a 'large' blockB ofC and that v B is concentrated around its expectation (see Lemmas 8.10 and 8.11).
3.2. Loops in the kernel. When studying the shortest and longest cycles in the kernel, the number of loops in the kernel plays a crucial role. We will prove in Section 7 that a typical kernel K on N edges has Θ (N ) many loops. Firstly, we will show that the kernel K (P ) 'behaves' asymptotically like a random cubic planar multigraph (see Lemma 7.1). Secondly, we estimate the typical number of loops in a random cubic planar multigraph by the second moment method (see Lemma 7.6). To this end, we introduce the so-called loop insertion (see Definition 7.2), which is a natural operation that changes an arbitrary cubic graph with 2n − 2 vertices to a cubic graph with 2n vertices and an additional loop. By using this loop insertion we can estimate the probability that there is a loop at some fixed vertex in a random cubic planar multigraph, from which we deduce the typical number of loops in a random cubic planar multigraph.
3.3. Blocks of the kernel. As in Section 3.2 we will use the fact that the kernel K (P ) behaves asymptotically like a random cubic planar multigraph. In order to analyse the block structure of a random connected cubic planar multigraph M , we assign the bridge number β (e) to a bridge e, defined as the order of the smaller component which we obtain from M by deleting e (see Definition 8.1). We will show that the bridge number β (e) is typically quite 'small': In fact, we will determine the distribution of a bridge number (see Lemma 8.4), using the so-called bridge insertion operation (see Definition 8.2). Then we combine it with a double counting argument to show that there is one block B 1 of linear order (see Lemma 8.10). For the second largest block B 2 in M we consider the maximum A of all bridge numbers. As there is a bridge e such that B 1 and B 2 lie in different components of M −e, we will get v (B 2 ) ≤ A. On the other hand, if e is a bridge with β (e) = A, then the smaller component of M − e is distributed similarly as a random connected cubic planar multigraph on A vertices. Hence, this smaller component should contain a block of linear order (in A). Thus, the second largest block (and by induction also the i -th largest block for every i ≥ 2) is of the same order as the maximum bridge number A.
3.4. Conditional random graphs. In Section 3.1 we considered a random core C = C (K , k) obtained from a (candidate) kernel K by randomly subdividing the edges of K by k additional vertices. In other words, given K and k, we considered the 'conditional' random core C conditioned on the event that its kernel is equal to K and its subdivision number is equal to k. However, we are actually interested in the 'unconditional' random core C (P ) of a random planar graph P = P (n, m) ∈ R P (n, m) for some function m = m(n). In this section we describe a method how to obtain results on an 'unconditional' random graph by studying the corresponding 'conditional' random graphs (see Lemma 3.2). To do so, we need the following definition.

Definition 3.1. Given a class A of graphs, a set S , and a function
Moreover, for each n ∈ N we denote by (A | s) (n) a graph chosen uniformly at random from the set {H ∈ A (n) : Φ(H ) = s n }. We will often omit the dependence on n and write just A | s (i.e. 'A conditioned on s') instead of (A | s) (n).

Lemma 3.2. Let A be a class of graphs, S a set, Φ : A → S a function, and Q a graph property. Let
The proof of Lemma 3.2 is provided in Section 10. In the following we illustrate how one can use Lemma 3.2 to deduce that a graph property Q holds whp in the core C (P ) of P = P (n, m) by studying the random core C (K , k), which is obtained by randomly subdividing the edges of a kernel K with k additional vertices. We start with a 'strong' property T that is whp satisfied by the kernel K (P ) and the subdivision number S (P ), e.g. v (K (P )) and S (P ) lie in certain intervals (see Theorem 4.8). Then we let A (n) ⊆ P (n, m) be the subclass of all graphs in P (n, m) fulfilling T and A := n∈N A (n). We define the function Φ : A → K × N by Φ(H ) := (K (H ) , S (H )) and let s = (K n , k n ) n∈N be a sequence that is feasible for (A , Φ). The core C (A | s) of the conditional random graph A | s is distributed like C (K n , k n ) (see Lemma 6.2 for details). Now the main step is to show that whp C (K n , k n ) ∈ Q. This is usually much easier than proving whp C (P ) ∈ Q directly, as the kernel and subdivision number are not random anymore in C (K n , k n ) and furthermore, fulfil property T . Knowing that whp C (K n , k n ) ∈ Q, it follows by Lemma 3.2 that whp also the core C (A) of the random graph A = A(n) ∈ R A (n) satisfies Q. Finally, this implies that whp C (P ) ∈ Q as desired, because whp P ∈ A by definition of A . Applications of Lemma 3.2 can be found e.g. in the proofs of Theorems 4.5 and 8.12.

KERNEL-STABLE CLASSES OF GRAPHS
In this section we will show that Theorem 1.4 holds for a more general class of graphs, called a kernel-stable class, in which graphs satisfy certain properties that are extracted from the class of planar graphs and are essential for the aforementioned core-kernel approach. Before defining the kernel-stable class, we first recall well-known classes of graphs (see e.g. [2,24,33,34] In addition, for each i ∈ N, the asymptotic probability that v (L 1 (K (2n, 3n))) = 2n − 2i is bounded away from both 0 and 1.
The constant α in (K2) is called the critical exponent. For short, we say P is kernel-stable with critical exponent α if it satisfies (P1) and (P2).
We can show that condition (K3) in Definition 4.1 can be deduced from conditions (K1) and (K2) if P is addable. In addition, each graph without complex components is in any kernel-stable class. The proofs of Lemmas 4.2 and 4.3 can be found in Section 10.

Lemma 4.3. Let P be a kernel-stable class of graphs and H a graph without complex components.
Then H ∈ P .
The next lemma indicates that a kernel-stable class is quite universal and rich, because it includes the class of cactus graphs (a cactus graph is a graph in which every edge belongs to at most one cycle), the class of series-parallel graphs, the class of planar graphs, and the class of graphs on a surface. (a) the class of cactus graphs with α = 5/2; (b) the class of series-parallel graphs with α = 5/2; (c) the class of planar graphs with α = 7/2; (d) the class of graphs embeddable on an orientable surface of genus g ∈ N with α = −5g /2 + 7/2.
Obviously, these classes also fulfil (P1), (K1) and are addable. Thus, they are kernel-stable classes due to Lemma 4.2. Moreover, in [23] it was shown that the class of graphs that are embeddable on an orientable surface of genus g ∈ N ∪ {0} satisfies (P2). Thus, they also form a kernel-stable class of graphs, since they trivially fulfil (P1).
Instead of proving Theorem 1.4 only for the class of planar graphs, we will show the following generalisation to kernel-stable classes of graphs in Section 9.

Theorem 4.5. Theorem 1.4 is true for any kernel-stable class of graphs.
This immediately implies that Theorem 1.4 is also true for the graph classes in Lemma 4.4. Corollary 4.6. Theorem 1.4 is true for the class of cactus graphs, the class of series-parallel graphs, and the class of graphs embeddable on an orientable surface of genus g ∈ N ∪ {0}.
In contrast to the classes of graphs in Corollary 4.6, the class of outerplanar graphs is not kernelstable, since subdividing an edge in an outerplanar graph can lead to a non-outerplanar graph. Hence, a non-outerplanar graph can have an outerplanar kernel, and thus (K1) is violated. Nevertheless, we can prove that Theorem 1.4 is also true for outerplanar graphs by using some results from [22]: (i) for the cases s 3 n −2 → −∞ and s = O n 2/3 we use that the asymptotic probability that the uniform random graph G(n, m) is outerplanar is bounded away from 0 (see Theorem 4.9(a)); (ii) if s 3 n −2 → ∞, we use the fact that a random outerplanar graph is whp a cactus graph [22,Theorem 4]. In order to prove Theorem 4.5 in Section 9, we will need the following two known facts. The first statement was shown in [23] by applying the core-kernel approach and provides useful information about a typical core and a typical kernel when s 3 n −2 → ∞. The later one deals with the cases s 3 n −2 → −∞ and s = O n 2/3 .

Theorem 4.8 ( [23]
). Let P be a kernel-stable class of graphs, P = P (n, m) ∈ R P (n, m), L 1 = L 1 (P ) the largest component of P , and R = R (P ) = P \L 1 . Assume m = n/2+ s for s = s(n) = o(n) and s 3 n −2 → ∞. Then the following hold: We note that the results in Theorem 4.8 were not explicitly proven in [23], but they immediately follow by combining Theorems 1.4(iii) and 5.4(iii), (v), (vi) and Corollaries 5.3 and 5.5 from [23]. Strictly speaking, the authors of [23] proved Theorem 4.8 only for the class of graphs embeddable on an orientable surface of genus g ∈ N ∪ {0}, but they pointed out that Theorem 4.8 generalises to the more general setting of kernel-stable graph classes (see [23,Remark 8.3]). (a) lim inf n→∞ P G has no complex component > 0; We note that if P is a kernel-stable class of graphs, then each graph without a complex component lies in P (see Lemma 4.3). Thus, Theorem 4.9(a) implies lim inf n→∞ P [G(n, m) ∈ P ] > 0 in the case of s ≤ Mn 2/3 . That will be useful in the proof of Theorem 4.5.

PÓLYA URN MODEL
In this section we present several useful results on the Pólya urn model introduced in Section 3.1.

Model with N colours.
Given N , k ∈ N, there are initially N balls of N distinct colours F 1 , . . . , F N in a urn. In each step we draw a ball uniformly at random from the urn. Then the drawn ball is returned to the urn along with an additional ball of the same colour. We repeat that procedure k times. For each i ∈ [N ] we denote by X i the number of drawn balls of colour F i at the end of the procedure (i.e. after k steps).
To derive bounds on the length of the shortest and longest cycle in the core (see Lemma 6.7), we need the following bounds on the minimum and maximum values of the total numbers X 1 , . . . , X f of drawn balls of the first f colours when N , k, and f are functions in n. Although we believe such results should be known, we could not find them in literature and therefore we include their proofs in Appendix A for completeness. (1). Then the following hold. (a) Another useful fact about the Pólya urn model (which will be used in the proof of Lemma 6.4(a)) is the following result on the distribution of X i , whose proof can be found in Appendix A.

Model with two colours.
Given b, w, k ∈ N, there are initially b black and w white balls in a urn. Then we draw k times uniformly at random a ball from the urn. In each step we return the drawn ball together with an additional ball of the same colour. Then we denote by X the number of drawn black balls at the end of the procedure, i.e. after k steps.

RANDOM CORE
In this section we study the process of obtaining a random core C = C (K , k) from a fixed kernel K by randomly subdividing the edges of K with k additional vertices for given (K , k). Because it is hard to directly analyse C , we circumvent this difficulty by introducing an auxiliary random multigraphC which behaves asymptotically like C and fits into the scheme of the Pólya urn model. Definition 6.1. Given a pair (K , k) of a multigraph K on vertex set [v (K )] of minimum degree at least three and k ∈ N, we denote by C (K , k) the set of all simple graphs on vertex set [v (K ) + k] obtained from K by subdividing the edges of K by the vertices v (K ) + 1, . . . , v (K ) + k. In other words, C (K , k) is the set of all cores on vertex set [v (K ) + k] whose kernel is equal to K . Let C = C (K , k) ∈ R C (K , k). In addition, we define a random multigraphC =C (K , k) by the following random experiment: we start with G 0 = K . Given the multigraph G i −1 we construct G i as follows (for i = 1, . . . , k). We choose uniformly at random an edge e of G i −1 and subdivide e by one additional vertex, which obtains the label v (K )+i . We note that E (G i −1 ) is a multiset, i.e. if there are r edges between vertices v and w , we choose one of these edges with probability r /e (G i −1 ). Then we letC = G k be the resulting multigraph after k steps.
Later we will study a random kernel-stable graph P conditioned on the event that K (P ) = K and S (P ) = k for some fixed kernel K and fixed k ∈ N. The next lemma says that the core of this 'conditional' random graph is distributed like C (K , k) from Definition 6.1. That fact will be quite useful when we apply Lemma 3.2. Lemma 6.2. Let P be a kernel-stable class of graphs and K the class of all kernels of graphs in P . Given a pair (K , k) of a multigraph K ∈ K on vertex set [v (K )] and k ∈ N, we let P (K ,k) (n, m) be the subclass of P (n, m) consisting of all graphs F whose kernel K (F ) is equal to K and whose subdivision number S(F ) is equal to k, i.e.
Let P | (K , k) ∈ R P (K ,k) (n, m) and C (K , k) be the random core as defined in Definition 6.1. Then the core of P | (K , k) is distributed like C (K , k): for each fixed graph H , we have In the next step, we provide a relation between the two random (multi-) graphs C andC (see Lemma 6.4). In particular, we show that if k = ω N 2 , they are contiguous in the sense of Definition 2.5 (see Corollary 6.5), where N = e (K ) is the number of edges in K . In our applications we will have k = Θ sn −1/3 and N = Θ sn −2/3 (see Theorem 4.8) and we note that sn −1/3 = ω sn −2/3 2 . In order to state this result, we need the following definition.
(b) Conditioning on the event thatC is simple, the distributions ofC and C are the same: for each graph H , we have Amongst others, Lemma 6.4(a) states that whpC is 2-simple if k = ω N 2 . Using that we can deduce the following result, which we will use later in the proof of Lemma 7.1. Corollary 6.6. Let P be a kernel-stable class of graphs and P = P (n, m) ∈ R P (n, m). Assume m = n/2 + s for s = s(n) = o(n) and s 3 n −2 → ∞. Then whp P is 2-simple.
Our next results provide bounds on the lengths g C and c C of the shortest and longest cycle iñ C . We note that if k = ω N 2 , these results also hold for C due to Corollary 6.5. Lemma 6.7. For every n ∈ N, let K = K (n) be a multigraph on vertex set [v (K )] of minimum degree at least three with N = N (n) edges and λ = λ(n) loops. In addition, let k = k(n) ∈ N and the random multigraphC =C (n) =C (K (n), k(n)) be as in Definition 6.1. We assume that N = ω (1). , In order to use Lemma 6.7 in the proof of Theorem 4.5, we will study the number of loops in a typical kernel of a random kernel-stable graph in Section 7 (see Corollary 7.7).
We conclude this section with a result on the block structure ofC , which provides also an insight into the block structure of C due to Corollary 6.5. Roughly speaking, the next lemma says that the i -th largest block of a cubic kernel K translates (during the process of constructingC ) to the i -th largest block ofC , provided the block is not too 'small'. Lemma 6.8. For every n ∈ N, let k = k(n) ∈ N, K = K (n) be a cubic multigraph on vertex set [v (K )], and the random multigraphC =C (n) =C (K (n), k(n)) be defined as in Definition 6.1. We assume that k = ω (v (K )) and let i ∈ N be fixed such that b i (K ) = ω (v (K )) 1/2 . Then whp Remark 6.9. The condition in Lemma 6.8 that b i (K ) is not too small, i.e. b i (K ) = ω (v (K )) 1/2 , can be weakened by using stronger concentration results on the Pólya urn model than the results in Theorem 5.3. We believe that using similar results as in Theorem 5.1 one can show that the statement of Lemma 6.8 is true even under the condition that b i (K ) = ω ((v (K )) ε ) for some ε > 0.

RANDOM KERNEL
Throughout this section, let P be a kernel-stable class of graphs and K the class of all kernels of graphs in P . We let P = P (n, m) ∈ R P (n, m) and consider the weakly supercritical regime when m = n/2 + s for s = s(n) = o(n) and s 3 n −2 → ∞.
Due to Theorem 4.8 we know that whp K (P ) is cubic and v (K (P )) = Θ sn −2/3 . Hence, we might expect that K (P ) 'behaves' asymptotically like a graph chosen uniformly at random from all cubic multigraphs in K with Θ sn −2/3 many vertices. In the next lemma we show that this is indeed true. We note, however, that this result is not straightforward, since K (P ) is not equally distributed on the set of all possible cubic kernels in K with Θ sn −2/3 many vertices. Lemma 7.1. Let P be a kernel-stable class of graphs and K the class of all kernels of graphs in P . Let F : P → N be a graph theoretic function and g 1 , g 2 : N → N non-decreasing functions. We assume that for K (2n, 3n) ∈ R K (2n, 3n), we have whp g 1 (n) ≤ F (K (2n, 3n)) ≤ g 2 (n). In addition, assume m = n/2 + s for s = s(n) = o(n) and s 3 n −2 → ∞. Then there exist constants c 2 ≥ c 1 > 0 such that for P = P (n, m) ∈ R P (n, m) we have whp g 1 c 1 sn −2/3 ≤ F (K (P )) ≤ g 2 c 2 sn −2/3 . We recall that λ (H ) denotes the number of loops in a multigraph H . Next, we will show that for K (2n, 3n) ∈ R K (2n, 3n), we have whp λ (K (2n, 3n)) = Θ(n), which implies whp λ (K (P )) = Θ sn −2/3 by Lemma 7.1. Our proof of that result will be based on the following observation. Let H be a cubic multigraph with a single loop at w . Then w has precisely one neighbour x = w . Assuming that there is no loop at x, there are two (not necessarily distinct) additional neighbours y and z of x. Then we obtain again a cubic multigraph if we delete w and x in H and add an edge y z. We note that by reversing this operation we can create a multigraph with a loop at w . This reverse operation leads to the following definition (see also Figure 2). Note that subdividing an edge which occurs r times in a multigraph H increases the weight of H by a factor of r . Therefore, we obtain the following. In order to see that Remark 7.5 is true, we imagine that we perform a loop insertion in two steps. Firstly, we choose the edge e ∈ E (H ) and subdivide it by the vertex n + 1. Secondly, we add the vertex n +2 together with an edge {n + 1, n + 2} and a loop at n +2. Then Remark 7.5 follows from Remark 7.4 and the fact that inserting a loop halves the weight of a graph.
Next, we show that typically the number of loops in the kernel is linear (in the number of edges in the kernel). Due to Lemma 7.1 it suffices to prove this result only for a random cubic kernel chosen from K (2n, 3n). We use a loop insertion to construct all graphs in K (2n, 3n) with a loop at a fixed vertex v. By doing that we can estimate the expected number of loops. Then we use the second moment method to show concentration around the mean. Recall that the number of loops in a graph H is denoted by λ (H ). Lemma 7.6. Let P be a kernel-stable class of graphs, K the class of all kernels of graphs in P , and K (2n, 3n) ∈ R K (2n, 3n). Then whp where γ > 0 is as in Definition 4.1.

BLOCK STRUCTURE
In this section we present several results which lead to the conclusion that Theorem 1.5 is also true for all kernel-stable classes of graphs P that are addable and have a critical exponent 3 < α < 4 (see Theorem 8.12). First we will analyse the block structure of a random cubic multigraph chosen from an appropriate class of multigraphs M (the so-called bridge-stable class), which will be later chosen as the class of connected kernels of graphs in P . Then we will deduce the block structure of the random kernel-stable graph in Theorem 8.12 by using Lemmas 6.8 and 7.1.
We recall that, as in the case of a simple graph, a block of a multigraph H is a maximal 2-connected subgraph of H . Here we insist that a vertex with a loop forms a block. In order to understand the The bridge number β (w x) in the graph on the right-hand side is 5.
block structure of a random cubic multigraph chosen from the class M , we will first analyse bridge numbers defined below (see also Figure 3). We observe that if w x ∉ E (H ), then we have β (w x) = 0. We will determine the distribution of the bridge number β (w x) for fixed vertices w and x. Intuitively, if β (w x) is typically 'small', then we should have a unique largest block which is significantly larger than all other blocks. We will show that this is indeed the case (see Lemmas 8.10 and 8.11). To do so, we introduce the bridge insertion, which is a operation that connects two components of a graph via a bridge (see also Figure 3). In order to successfully apply bridge insertions in some graph class M , we require two natural properties of M . having 3n edges. Then there exist constants γ > 0, c > 0, and α ∈ R such that The constant α in (B2) is called the critical exponent.
We note that the class of all connected kernels of an addable kernel-stable class is bridge-stable. Next, let M be some bridge-stable class of multigraphs. We study the asymptotic distribution of the bridge number β M (v w ) for fixed vertices v = w and M = M (2n, 3n) ∈ R M (2n, 3n). More precisely, we estimate the probability P β M (w x) ≥ 2 f (n) + 1 for an arbitrary function f .  (2n, 3n) and let c and γ be as in (6). Then for any pair of distinct vertices w, x ∈ [2n] we have Using the second moment method, we will show that this number is concentrated around its expectation.

Lemma 8.5. Let M be a bridge-stable class of multigraphs with critical exponent α > 3 and f be a function with f (n) = ω(1), f (n) = o(n), and f (n) = o(n 1/(α−2)
). Let M = M (2n, 3n) ∈ R M (2n, 3n). In addition, let w 1 , w 2 , w 3 , w 4 ∈ [2n] be pairwise distinct and let c and γ be as in (6). Then we have  (2n, 3n) and let c and γ be as in (6). Then whp M has Next, we show in two steps that whp M = M (2n, 3n) ∈ R M (2n, 3n) has a block with linearly many vertices. Firstly, we prove that whp M has a dominant block (see Definition 8.7 and Lemma 8.8). Secondly, we show that whp this dominant block has linearly many vertices (see Lemma 8.10).
Assuming that the bridge numbers β M (e i ) are 'small', we get that v (B ) or r needs to be 'large'. We note that in the latter case we again obtain that v (B ) is 'large', since each vertex in B can lie in at most one bridge. In Lemma 8.10 we will make this idea more precise. We already saw in Lemma 8.4 that the bridge numbers are typically 'small'. However, we need the following stronger result in the proof of Lemma 8.10.  Lemmas 8.10 and 8.11 together with Lemma 7.1 give us the block structure of a random kernel K (P ). Now we combine this information with Lemma 6.8 to obtain the block structure of a random kernel-stable graph. Theorem 8.12. Let P be a kernel-stable class of graphs which is addable and has a critical exponent 3 < α < 4. In addition, let P = P (n, m) ∈ R P (n, m) and L 1 = L 1 (P ) denote the largest component of P . Assume m = n/2 + s for s = s(n) = o (n) and s 3 n −2 → ∞. Then the following hold.
(c) The number of blocks in L 1 is whp Θ sn −2/3 .
The proof of Theorem 8.12 can be found in Section 9.5.
Remark 8.13. We believe that Theorem 8.12 is also true when α ≥ 4. In that case we would need an improved version of Lemma 6.8 where we can weaken the condition that b i (K ) = ω (v (K )) 1/2 to b i (K ) = ω ((v (K )) ε ) for some ε > 0. Using the ideas presented in Section 5 one may deduce such an improved statement (see also Remark 6.9). Nevertheless, we omit details, since we expect that the proofs become rather technical but do not provide any new insights.
Next, we will apply Lemma 3.2 to the class A := n∈N A (n). So, we define the function Φ : A → K × N which maps a graph H ∈ A to the pair of kernel K (L 1 (H )) and subdivision number S (L 1 (H )), i.e.
Φ(H ) := (K (L 1 (H )) , S (L 1 (H ))) . Let s = (K n , k n ) n∈N be a sequence that is feasible for (A , Φ) (cf. Definition 3.1) and let A = A(n) ∈ R A (n). Due to the definition of Φ all possible realisations of L 1 (A | s) have the same kernel K n and the same subdivision number k n . Hence, by Lemma 6.2 we have that C (L 1 (A | s)) is distributed like C (K n , k n ), a graph chosen uniformly at random from the class of all cores with kernel K n and subdivision number k n . From the definition of A (n) we have k n = Θ sn −1/3 , e (K n ) = Θ sn −2/3 , and λ (K n ) = Θ sn −2/3 . In particular, this yields k n = ω e (K n ) 2 and λ (K n ) = Θ (e (K n )). Hence, by combining Lemma 6.7 and Corollary 6.5 we obtain g (L 1 (A | s)) = g (C (K n , k n )) = Θ p k n e (K n ) 2 = Θ p sn −1/3 sn −2/3 2 = Θ p (n/s) and c (L 1 (A | s)) = c (C (K n , k n )) = Ω p k n e (K n ) log λ (K n ) = Ω p n 1/3 log sn −2/3 .
As the sequence s was arbitrary, Lemma 3.2 implies that the results above hold also for the (unconditional) random graph A. Since whp P ∈ A (n), the same is true for P , i.e.
We setÃ := n∈NÃ (n) and let δ > 0. By Theorem 4.8(f) and (g) we can choose the constant M such that We claim that each graph R ∈ G (n U , m U ) having no complex components and satisfying L 1 (R) < v (H n ) is in R(n). Indeed such a graph R satisfies L 1 (R ∪ H n ) = H n . Moreover, we have K (R ∪ H n ) = K (H n ). Thus, by the stability property (K1) of kernel-stable classes (cf. Definition 4.1) we get R ∪H n ∈ P and therefore R ∪ H n ∈Ã . This implies R ∈ R(n) due to (9). Next, we will show that |R| is 'large' in the sense that |R(n)|/ |G (n U , m U )| is bounded away from 0. To this end, we use (7) to obtain Using (7) and the fact that s = o(n) we get n U = (1 + o(1))n. Combining that with (10) yields that for n large enough Together with Theorem 4.9(a) this implies lim inf n→∞ P R n has no complex component > 0. (12) As v (H n ) = Θ(s), we obtain by Theorem 4.9(b) that whp Combining (12) and (13) with the claim shown above yields Similarly as in (10) we use (7) to get which yields m U ≥ n U /2 − 2Mn 2/3 U for large n. Combining that with (11) we obtain m U = n U /2 + O n 2/3 U . Hence, we get by Theorem 1.2(b) that c (R n ) = Θ p n 1/3 U = Θ p n 1/3 . By (14) each property that holds whp in R n is also true whp inR n ∈ R R(n). Thus, we have c R n = Θ p n 1/3 . From the definition ofR n we get c R Ã | H = Θ p n 1/3 . Hence, by Lemma 3.2 we have c R Ã = Θ p n 1/3 . Finally, using (8) and observing that the choice of δ > 0 was arbitrary we have c (R (P )) = Θ p n 1/3 , which completes the proof. The results in the regimes s 3 n −2 → −∞ and s = O n 2/3 follow analogously as for kernel-stable classes (see Section 9.1). In [22,Theorem 4] it is proven that in the case s 3 n −2 → ∞ whp a random outerplanar graph is a cactus graph. Thus, the statements in that regime follow directly from Corollary 4.6. 9.5. Proof of Theorem 8.12. We start by considering the blocks of the kernel K (P ). By Lemma 7.1 we have that K (P ) 'behaves' like a random cubic multigraph chosen from K . Furthermore, by Theorem 4.8(b) and (d), we know that whp v (K (P )) = Θ sn −2/3 . Combining that with Lemma 8.10 implies Similarly, by Lemma 8.11 we have that for each i ≥ 2 b i (K (P )) = Θ p v (K (P )) 1/(α−2) = Θ p s 1/(α−2) n −2/(3(α−2)) .
Next, we determine the orders of the blocks in the core C (P ). To this end, we will use Lemmas 3. Let s = (K n , k n ) n∈N be a sequence feasible for (A , Φ) and C n := C (K n , k n ) as in Definition 6.1, i.e. a graph chosen uniformly at random from all cores with kernel K n and subdivision number k n . By the definition of A (n) we have k n = ω e (K n ) 2 . Thus, combining Corollary 6.5 and Lemma 6.8 yields Due to Lemma 6.2 we have that C (A | s) is distributed like C n . Hence, by Lemma 3.2 we obtain that whp We note that each block outside the core C (P ) is a cycle. Due to Theorem 4.5, the length of such a cycle is of order O p n 1/3 . This together with (17) and the observations that implies statements (a) and (b).
For statement (c), we first observe that the number of blocks in C (L 1 (P )) is at most v (K (L 1 (P ))). Thus, by Theorem 4.8(a) we have that whp C (L 1 (P )), and therefore also L 1 (P ), has O sn −2/3 many blocks. On the other hand, whp K (L 1 (P )) has Θ sn −2/3 many loops due to Corollary 7.7. All these loops 'translate' to different blocks in the core C (L 1 (P )). Thus, whp L 1 (P ) has Ω sn −2/3 many blocks. Summing up, whp L 1 (P ) contains Θ sn −2/3 many blocks. . , e N be the edges of K and we denote by X i the number of vertices that are placed on edge e i when we subdivide K to obtainC . To prove (a), we observe that where the last inequality follows from Proposition 5.2. In order to prove (b), it suffices to show that P C = H is independent of the choice of H ∈ C (K , k). To that end, we count the number of ways our random process ends up withC = H . We observe that there is a unique sequence (G 0 , . . . ,G k ) that leads to G k = H . Thus, in each step i there is a unique unordered pair of vertices {u i , v i } such that subdividing an edge between u i and v i in G i −1 leads to G i . We denote by q i the number of edges in G i −1 between u i and v i . Then, there are k i =1 q i many ways of creating H . We note that the only way a multiple edge can be created during the process is by subdividing a loop and that all loops and multiple edges are destroyed in the end. Thus, we obtain (2). This shows (b), since w (K ) is independent of the choice of H ∈ C (K , k).
10.6. Proof of Corollary 6.5. We observe that by Lemma 6.4 we have that whpC is simple. For each graph property Q we obtain by Lemma 6.4(b) This implies that whp C ∈ Q if and only if whpC ∈ Q. 10.7. Proof of Corollary 6.6. We will use Lemma 3.2. Let Q be the graph property of being 2-simple. In addition, let A (n) be the subclass of P (n, m) consisting of all graphs H with v (C (H )) = Θ sn −1/3 and e (K (H )) = Θ sn −2/3 . Due to Theorem 4.8(a) and (c) we have that whp v (C (P )) = Θ sn −1/3 . In addition, by Theorem 4.8(b), (d), and (e) we have that whp e (K (P )) = 3/2 · v (K (P )) = Θ sn −2/3 . Hence, we obtain that whp P ∈ A (n). Let A := n∈N A (n) and define the function Φ for a graph H ∈ A by Φ(H ) := (K (H ), S (H )). Let s = (K n , k n ) n∈N be a sequence feasible for (A , Φ) and let C (K n , k n ) andC (K n , k n ) be as in Definition 6.1. By definition of A (n) we have k n = ω e (K n ) 2 . Thus, by Lemma 6.4(a) we have that whpC (K n , k n ) is 2-simple. By Corollary 6.5 this is also true for C (K n , k n ). Let A = A(n) ∈ R A (n). We note that C (A | s) is distributed like C (K n , k n ) due to Lemma 6.2. Thus, by Lemma 3.2, we have that whp C (A) is 2-simple. Since whp P ∈ A (n), it is also true that whp C (P ) is 2-simple. Finally, the statement follows, because P is 2-simple if and only if C (P ) is.
10.8. Proof of Lemma 6.7. Let e 1 , . . . , e N be the edges of K and X i the number of vertices that are placed on edge e i if we subdivide K to obtainC . Without loss of generality we may assume that e 1 , . . . , e λ are the loops of K . Then the upper bounds on g C follow by Theorem 5.1(a) and inequality (3). For the lower bound on g C we use Theorem 5.1(a) and (4). The 'in particular' statements follow immediately by combining the lower and upper bounds on g C . Finally, we note that (b) follows by Theorem 5.1(b) and (5).
10.9. Proof of Lemma 6.8. We note that the blocks ofC are the blocks of K with additional vertices placed on the edges of K . For j ∈ N let X j be the total number of vertices that are placed on edges of the j -th largest block B j (K ) of K . The minimum degree of each block is at least two and together with the fact that K is cubic, this implies By Theorem 5.3 we have Thus, by Chebyshev's inequality, (18), and (19) we obtain Hence, whp for all j ≤ i we have X j ≥ v(K ) . By again applying Chebyshev's inequality, (18), and (19), we have uniformly over all j ≥ i Thus, by a standard union bound we obtain , which completes the proof. 10.10. Proof of Lemma 7.1. We will use Lemma 3.2. To this end, let c 1 , c 2 > 0 and A (n) be the subclass of P (n, m) consisting of all 2-simple graphs H with a cubic kernel K (H ) and satisfying c 1 sn −2/3 ≤ v (K (H )) /2 ≤ c 2 sn −2/3 . Due to Corollary 6.6 we know that whp P is 2-simple. Moreover, by Theorem 4.8(b), (d), and (e) we have that whp K (P ) is cubic and v (K (P )) = Θ sn −2/3 . Thus, we can choose c 1 , c 2 such that whp P ∈ A (n). Let A := n∈N A (n) and define the function Φ for a graph H ∈ A by Φ(H ) := v (K (H )) /2. Let s = (ℓ n ) n∈N be a sequence feasible for (A , Φ) and A = A(n) ∈ R A (n). We note that for a fixed kernel K ∈ K (2ℓ, 3ℓ) and a fixed k ∈ N ∪ {0} there are w (K ) k−3ℓ−1 3ℓ−1 k! many ways to construct a 2simple core with kernel K and subdivision number k. Thus, K (A | s) is distributed like K (2ℓ n , 3ℓ n ) by Lemma 6.2. Hence, we obtain P g 1 c 1 sn −2/3 ≤ F (K (A | s)) ≤ g 2 c 2 sn −2/3 ≥ P g 1 (ℓ n ) ≤ F (K (2ℓ n , 3ℓ n )) ≤ g 2 (ℓ n ) as ℓ n ≥ c 1 sn −2/3 → ∞. Thus, the statement follows by Lemma 3.2.
10.11. Proof of Lemma 7.6. We recall that λ (K ) is the number of loops in K = K (2n, 3n) and observe that λ (K ) = w∈[2n] Z w , where Z w is the indicator random variable for the event that there is a loop at vertex w . In order to apply the second moment method, we estimate the probabilities P [Z w = 1] and P [Z u = Z w = 1] for u = w . To this end, we will use loop insertions (cf. Definition 7.2). We fix a vertex w ∈ [2n] and consider all multigraphs in K (2n, 3n) with a loop at w . We note that in all these multigraphs w has precisely one neighbour x = w . We distinguish two cases depending on whether there is a loop at x or not. Due to Remark 7.3 we can enumerate all these multigraphs with no loop at x as follows: For simplicity we set a n := |K (2n, 3n)|. If x and H are fixed, the total weight of all multigraphs that can be built by choosing an edge e ∈ E (H ) and performing a loop insertion at edge e with vertex pair (w, x) is w (H ) 3(n − 1)/2 by Remark 7.5. Hence, the total weight of all multigraphs which can be obtained by the above construction is (2n − 1) · a n−1 · 3(n − 1)/2.
On the other hand, if there is a loop at x, the vertices w and x form a component with weight 1/4. Thus, all such multigraphs can be enumerated as follows: • choose a vertex x ∈ [2n] \ {w }; • choose H ∈ K (2(n − 1), 3(n − 1)) and relabel the vertices with the labels [2n] \ {w, x}; • add the component C with V (C ) = {w, x} and E (C ) = {w w, xx, w x} to H . The total weight of all multigraphs constructed in that way is (2n − 1)a n−1 /4.
By using (K2) in Definition 4.1 we obtain a n−1 a n = Plugging in (23) in (22) yields Similarly, we estimate the number of multigraphs with loops at u and w . We observe that all such multigraphs in which u and w are not adjacent can be construct as follows: • choose a vertex x ∈ [2n] \ {u, w }; • choose H ∈ K (2(n − 1), 3(n − 1)) and relabel the vertices with the labels [2n] \ {w, x} such that we obtain a multigraph with a loop at u; • choose an edge e = uu in H and perform a loop insertion at edge e with vertex pair (w, x) .
In the above construction we have (2n − 2) possible choices for x and by (24) the weight of all multigraphs that can be chosen for H is For fixed x and H , the total weight of all multigraphs obtained by choosing an edge e and performing the loop insertion is w (H )(3n − 4)/2 due to Remark 7.5. On the other hand, if u and w are adjacent and there are loops at u and w , then u and w form an own component with weight 1/4. Combining these two cases we get (1)) 3 4γ a n−1 3n − 4 2 + 1 4 a n−1 /a n = (1 + o(1)) 9n 2 4γ a n−1 a n .
Finally, from this together with (23) we obtain Hence, the statement follows by the second moment method.
In order to prove Lemmas 8.4 and 8.5, we need the following two results, whose proofs are elementary and can be found in Appendix B. By Remark 7.4 this construction gives multigraphs with a total weight of ( j ,k)∈I (n) where Hence, using Claim 1 yields as desired.
10.14. Proof of Lemma 8.5. We denote by E i the event that β (w 2i −1 w 2i ) ≥ 2 f (n) + 1, where i ∈ {1, 2}. In addition, let E 3 be the event that there is an edge in M with one endpoint in {w 1 , w 2 } and the other in {w 3 , w 4 }. We start by estimating the probability P [E 1 ∧ E 2 ∧ E 3 ]. We observe that if E 3 is true, then at least one of the four events w 1 w 3 ∈ E (M ), w 1 w 4 ∈ E (M ) , w 2 w 3 ∈ E (M ), w 2 w 4 ∈ E (M ) is true. Thus, by symmetry reasons we obtain Next, we note that the event E 2 implies w 3 w 4 ∈ E (M ). Using that in (26) yields Using Lemma 8.4 for an estimate of P [E 1 ] and the fact that Next, we estimate the probability P [E 1 ∧ E 2 ∧ ¬E 3 ], where ¬E 3 is the event that E 3 is not true. We observe that we can enumerate all multigraphs H ∈ M (2n, 3n) satisfying β H (w 1 w 2 ) , β H (w 3 w 4 ) ≥ 2 f (n) + 1, and w 1 w 3 , w 1 w 4 , w 2 w 3 , w 2 w 4 ∉ E (H ) by the following construction: • choose j , k, l ∈ N with j + k + l = n − 2 and j , k ≥ f (n); Hence, letting I (n) := j , k, l ∈ N 3 | j + k + l = n − 2, j , k ≥ f (n) , m i := |M (2i , 3i )| for i ∈ N, we obtain where the last equality follows from Claim 2.
10.15. Proof of Lemma 8.6. We will use the second moment method. To that end, let Z p be the indicator random variable that an unordered pair p = {w, x} of vertices satisfies β (w x) ≥ 2 f (n) + 1 and Z = p Z p . Now let w, x, y ∈ [2n] be distinct vertices and p = {w, x} and q = w, y be unordered pairs. Then we have Hence, by using Lemmas 8.4 and 8.5 we obtain which implies the statement due to the second moment method.
where we used in the last inequality the assumption that each bridge number is at most (2n − 2)/3. This implies that v corresponds to a block in H . As all bridges are oriented away from v, this block is dominant.
Then by using Jensen's inequality for the convex function x → x µ and the simple fact r ≤ v (B ), we obtain that for n large enough a contradiction. Hence, we obtain v (B ) ≥ n/h(n), which completes the proof.
Remark 10.1. In the proof of Lemma 8.10 we actually showed the following stronger statement. If there are Θ p (n) many vertices outside the largest block of M , then M contains a block which shares a vertex with Θ p (n) many bridges. We note that in most of our applications this assumption is satisfied, for example, when whp M contains linearly many loops (see Lemma 7.6).  (H , e), where H ∈ M (2n, 3n) and β H (e) ≥ L ′ (n). Moreover, let M ′ , e ′ be a pair chosen uniformly at random from M ′ (n). Next, we show that the distributions of M and M ′ are 'similar'. More precisely, let Q be some graph property. We claim that To prove it, we define β H , j := e ∈ E (H ) | β H (e) ≥ j for a multigraph H and j ∈ N. With this notation we obtain by Lemma 8.4 Next, we observe that for each H ∈ M (2n, 3n) we have Combining (28) and (29) yields where we assumed that whp M ′ ∈ Q. Finally, we observe that for each δ > 0, we can choose L > 0 such that P β M , L ′ = 0 ≤ δ by Lemma 8.4. This shows (27). Next, we prove b i M ′ = Ω p n 1/(α−2) , which implies b i (M ) = Ω p n 1/(α−2) by (27). We note that we can enumerate all pairs (H , e) ∈ M ′ (n) as follows: • choose an unordered pair {w, x} with w = x ∈ [2n] and set e = w x; • choose j , k ∈ N such that j + k = n − 1 and j , k ≥ L ′ /2; • choose a partition J∪ K = [2n] \ {w, x} with |J | = 2 j and |K | = 2k; • choose H 1 ∈ M (2 j , 3 j ) and H 2 ∈ M (2k, 3k) and relabel the vertices with J and K , respectively; • choose edges e 1 ∈ E (H 1 ) and e 2 ∈ E (H 2 ) and perform a bridge insertion (cf. Definition 8.2) at edges e 1 and e 2 with vertices w and x.
Next, we will use Lemma 3.2, which we only formulated for graph classes, but it is straightforward that it is also true for classes of graphs where one edge is marked. Let A (n) = M ′ (n) and define the function Φ for a pair (H , e) ∈ A := n∈N A (n) by Φ(H , e) := e, β H (e) . Let e n , j n n∈N be a sequence feasible for (A , Φ) and (F n , e n ) be a pair chosen uniformly at random from all elements in A (n) that evaluate to e n , j n under Φ. Now let k n = n − 1 − j n and let C 1 and C 2 be the two graphs obtained by reversing the bridge insertion of e n in F n . By the above construction, C 1 and C 2 are distributed like M (2 j n , 3 j n ) and M (2k n , 3k n ), respectively. Without loss of generality we may assume j n ≥ (n − 1)/2. Now by Lemma 8.10 and induction hypothesis we obtain b i −1 (C 1 ) = Ω p n 1/(α−2) and b 1 (C 2 ) = Ω p (k n ) = Ω p n 1/(α−2) . Hence, we have b i (F n ) = Ω p n 1/(α−2) . Now we get b i M ′ = Ω p n 1/(α−2) by Lemma 3.2. This together with (27) implies that b i (M ) = Ω p n 1/(α−2) .

DISCUSSION
Theorem 1.4(c) raises the question about the precise asymptotic order of the circumference c (L 1 ) of the largest component L 1 of a uniform random planar graph in the weakly supercritical regime. The reason why we provided only a lower and an upper bound for c (L 1 ) is partly because we could not determine the precise order of the circumference c (K ) of a random cubic planar multigraph K . If there were a function f = f (n) such that c (K ) = Θ p ( f ), then our proof would imply that c (L 1 ) = Ω p n 1/3 f sn −2/3 and c (L 1 ) = O p n 1/3 f sn −2/3 log n . That closes the gap up to a factor of log n.
Moreover, if f (n) = ω n 1/2 , then our methods lead even to c (L 1 ) = Θ p n 1/3 f sn −2/3 . We note that Robinson and Wormald [39] showed that whp a random cubic (general, not necessarily planar) graph has a Hamiltonian cycle. We know that this is not the case for random cubic planar multigraphs, since the longest cycle misses all vertices which have a loop attached. By Lemma 7.6 there are linearly many such vertices. Nevertheless, we believe that whp there is a cycle of linear length.
Conjecture 11.1. Let K = K (2n, 3n) be a graph chosen uniformly at random from the class of all cubic planar multigraphs on vertex set [2n]. Then, we have c (K ) = Θ p (n).
If the above conjecture were true, we would immediately obtain the following result. In Theorem 8.12 we determined the block structure for kernel-stable classes of graphs which are addable and have a critical exponent 3 < α < 4. We already pointed out that it should be straightforward to generalise these results to the case α ≥ 4 (see Remark 8.13). However, we believe that this is not the case any more if α < 3. Panagiotou and Steger [38] showed that in random n-vertex graphs, there is a drastic change in the block structure when the critical exponent α is around 3. For example, they showed that a random planar graph on n vertices (whose critical exponent is known to be α = 7/2) has whp a block of order linear in n, while the largest block of a random outerplanar graph or in a random series-parallel graph on n vertices (whose critical exponent is known to be α = 5/2) is of order O log n . This leads to the following conjecture (see Lemmas 8.10 and 8.11 for comparable results for the case α > 3). If this were true, we would obtain the following result (see Theorem 8.12 for a comparable statement for α > 3). Before providing the proof of Theorem 5.1 we briefly illustrate the proof strategy. As for an upper bound for X * , we will find a 'small' function g 1 = g 1 (n) such that P X * ≥ g 1 = o (1). For 1 ≤ i ≤ f we denote by A i the event that X i ≥ g 1 and observe that X * ≥ g 1 if and only if A i is true for all 1 ≤ i ≤ f . Moreover, we intuitively expect that because given that X 1 , . . . , X i −1 are 'large' (for some 1 ≤ i ≤ f ), the probability that X i is also 'large' might decrease. If (30) holds (see Proposition A.2), then we obtain In order to derive a lower bound for X * , we will determine a 'large' function g 2 = g 2 (n) such that P X * ≤ g 2 = o (1). To that end, we observe that if X * ≤ g 2 , then there is at least one 1 ≤ i ≤ f such that X i ≤ g 2 . Thus, we obtain Therefore, in both cases, it is enough to find good bounds for P X i ≥ g 1 and P X i ≤ g 2 . Such bounds are obtained in Proposition A. 3.
In order to make the aforementioned idea more precise, we need two known facts about the Pólya urn model. The first one is about the marginal distribution of X i and will be our starting point for deducing bounds on P X i ≥ g 1 and P X i ≤ g 2 .
and P i j =1 We note that a more general version of Proposition A.2 was proven in [4, Example 5.5] by using a fact from [25, (1.8)]. A random vector (X 1 , . . . , X N ) satisfying (31) and (32) is also called negatively dependent (see e.g. [3] for details). Next, we derive some bounds for P [X i ≤ x] and P [X i ≥ x] by using Proposition A.1. k + N x .
(b) If in addition x ≤ k 2 , then we have (c) If in addition k ≥ 8N and x ≤ k 2 , then we have A.1. Proof of Proposition A.3. Throughout the proof, we use Proposition A.1 without stating explicitly. Then the first inequality in (a) follows by For the second inequality in (a) we may assume N ≥ 3, since otherwise the statement is trivially fulfilled. We get by using 1 + z ≤ exp(z) for z ∈ R Next, we observe that for y ∈ {0, . . . , k − 1} Hence, we obtain which proves (a). Next, we assume x ≤ k 2 and show (b). To that end, we use 1 − z ≥ exp(−2z) for z ∈ 0, 1 2 to obtain Using that yields This shows (b). Finally, we assume k ≥ 8N and x ≤ k 2 . Then, we have for y ≤ 3k 4 P X i = y + 1 Thus, for x ≤ k 2 we obtain by using (33) Hence, we conclude the proof with A.2. Proof of Theorem 5.1. Throughout the proof, we let n be large and h = h(n) = ω (1). To obtain the claimed bounds on X * = min 1≤i ≤ f X i , it suffices to show that as desired.
To prove (b), it suffices to show P X * ≥ hk N f = o(1) for any h = o f . Now let x = hk N f and for each 1 ≤ i ≤ f we denote by A i the event that X i ≥ x. If X * ≥ x, then X i ≥ x for all 1 ≤ i ≤ f . Thus, by Proposition A.2 we have By Proposition A.3(b), uniformly over all 1 ≤ i ≤ f , we have In order to derive the claimed bounds on X * = max 1≤i ≤ f X i , we prove the following assertions. To show (d), we assume k = ω(N ) and f = ω (1) and let x = k hN 1 + log f . If X * ≤ x, then X i ≤ x for all 1 ≤ i ≤ f . For each 1 ≤ i ≤ f we denote by B i the event that X i ≤ x. Using Proposition A.2 (for the first inequality) and Proposition A.3(c) (for the second inequality) yields for large n In order to prove (e), we let x = k hN and use Proposition A.3(a) to get where we used in the last equality that h = ω(1) and k = ω(N ).
To show (f), we assume k = ω(N ) and let x = hk N 1 + log f . If X * ≤ x, then X i ≤ x for some 1 ≤ i ≤ f . Therefore, by Proposition A.3(a) we obtain To prove (g), we assume k = O(N ) and let x = h 1 + log f . Using Proposition A.3(a) we get This concludes the proof.
In addition, for i ∈ {2, 3, 4} we set Similarly as in the proof of (35), we have Next, we observe S 3 ≤ S 2 and Combining that with (37) yields the statement.