Random planar maps and graphs with minimum degree two and three

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order c log(n) for an explicit constant c. These results provide new information on the structure of random planar graphs.


Introduction
The main goal of this paper is to enumerate planar graphs subject to a condition on the minimum degree δ, and to analyze the corresponding planar random graphs. Asking for δ ≥ 1 is not very interesting, since a random planar graph contains in expectation a constant number of isolated vertices. The condition δ ≥ 2 is directly related to the concept of the core of a graph. Given a connected graph G, its core (also called 2-core in the literature) is the maximum subgraph C with minimum degree at least two. The core C is obtained from G by repeatedly removing vertices of degree one. Conversely, G is obtained by attaching rooted trees at the vertices of C. The kernel of G is obtained by replacing each maximal path of vertices of degree two in the core C with a single edge. The kernel has minimum degree at least three, and C can be recovered from K by replacing edges with paths. Notice that G is planar if and only C is planar, if and only if K is planar.
As shown in Figure 1, the kernel may have loops and multiple edges, which must be taken into account since our goal is to analyze simple graphs. Another issue is that when replacing loops and multiple edge with paths the same graph can be produced several times. To this end we weight multigraphs appropriately according to the number of loops and edges of each multiplicity. We remark that the concepts of core and kernel of a graph are instrumental in the theory of random graphs [13,15].
It is convenient to introduce the following definitions: a 2-graph is a connected graph with minimum degree at least two, and a 3-graph is a connected graph with minimum degree at least three. In order to enumerate planar 2-and 3-graphs, we use generating functions. From now on all graphs are labelled and generating functions are of the exponential type. Let c n , h n and k n be, respectively, the number of planar connected graphs, 2-graphs and 3-graphs with n vertices, and let C(x) = c n x n n! , H(x) = h n x n n! , K(x) = k n x n n! be the associated generating functions. Also, let t n = n n−1 be the number of (labelled) rooted trees with n vertices and let T (x) = t n x n /n!. The decomposition of a connected graph into its core and the attached trees implies the following equation where U (x) = T (x) − T (x) 2 /2 is the generating functions of unrooted trees. Since T (x) = xe T (x) , we can invert the above relation and obtain The equation defining K(x) is more involved and requires the bivariate generating function C(x, y) = c n,k y k x n n! , where c n,k is the number of connected planar graphs with n vertices and k edges. We can express K(x) in terms of C(x, y) as K(x) = C(A(x), B(x)) + E(x), where A(x), B(x), E(x) are explicit elementary functions (see Section 3). From the expression of C(x) as the solution of a system of functional-differential equations [10], it was shown that c n ∼ κn −7/2 γ n n!, where κ ≈ 0.4104 · 10 −5 and γ ≈ 27.2269 are computable constants. In addition, analyzing the bivariate generating function C(x, y) it is possible to obtain results on the number of edges and other basic parameters in random planar graphs. Our main goal is to extend these results to planar 2-graphs and 3-graphs. Using Equations (1) and (2) we obtain precise asymptotic estimates for the number of planar 2-and 3-graphs: h n ∼ κ 2 n −7/2 γ n 2 n!, γ 2 ≈ 26.2076, κ 2 ≈ 0.3724 · 10 −5 , k n ∼ κ 3 n −7/2 γ n 3 n!, γ 2 ≈ 21.3102, κ 3 ≈ 0.3107 · 10 −5 .
As is natural to expect, h n and k n are exponentially smaller than c n . Also, the number of 2connected planar graphs is known to be asymptotically κ c n −7/2 26.1841 n n! (see [3]), smaller than the number of 2-graphs. This is consistent, since a 2-connected has minimum degree at least two. By enriching Equations (1) and (2) taking into account the number of edges, we prove that the number of edges in random planar 2-graphs and 3-graphs are both asymptotically normal with linear expectation and variance. The expected number of edges in connected planar graphs was shown to be [10] asymptotically µn, where µ ≈ 2.2133. We show that the corresponding constants for planar 2-graphs and 3-graphs are µ 2 ≈ 2.2614, µ 3 ≈ 2.4065.
This conforms to our intuition that increasing the minimum degree also increases the expected number of edges. We also analyze the size X n of the core in a random connected planar graph, and the size Y n of the kernel in a random planar 2-graph. We show that both variables are asymptotically normal with linear expectation and variance and that E X n ∼ λ 2 n, λ 2 ≈ 0.9618, E Y n ∼ λ 3 n, λ 3 ≈ 0.8259.
We remark that the value of λ 2 has been recently found by McDiarmid [14] using alternative methods. Also, we remark that the expected size of the largest block (2-connected component) in random connected planar graphs is asymptotically 0.9598n [11]. Again this is consistent since the largest block is contained in the core. The picture is completed by analyzing the size of the trees attached to the core. We show that the number of trees with k vertices attached to the core is asymptotically normal with linear expectation and variance. The expected value is asymptotically where C > 0 is a constant and ρ ≈ 0.03673 is the radius of convergence of C(x). For k large, the previous quantity grows like This quantity is negligible when k ≫ log(n)/(log(1/ρe)). Using the method of moments, we show that the size L n of the largest tree attached to the core is in fact asymptotically log(n) log(1/ρe) .
Moreover, we show that L n / log n converges in law to a Gumbel distribution. This result provides new structural information on the structure of random planar graphs. Our last result concerns the distribution of the vertex degrees in random planar 2-graphs and 3-graphs. We show that for each fixed k ≥ 2 the probability that a random vertex has degree k in a random planar 2-graph tends to a positive constant d H (k), and for each fixed k ≥ 3 the probability that a random vertex has degree k in a random planar 3-graph tends to a positive constant d K (k). Moreover k≥2 p H (k) = k≥3 p K (k) = 1, and the probability generating functions are computable in terms of the probability generating function p C (w) of connected planar graphs, which was fully determined in [6].
The previous results show that almost all planar 2-graphs have a vertex of degree two, and almost all planar 3-graphs have a vertex of degree three. Hence asymptotically all our results hold also for planar graphs with minimum degree exactly two and three, respectively. In addition, all the results for connected planar graphs extend easily to arbitrary planar graphs. This is because the expected size of the largest component in a random planar graph is n − O(1) (see [11]). We will not repeat for each of our results the corresponding statement for graphs of minimum degree exactly two or three.
It is natural to ask why we stop at minimum degree three. The reason is that there seems to be no combinatorial decomposition allowing to deal with planar graphs of minimum degree four or five (a planar graph has always a vertex of degree at most five). It is already an open problem to enumerate 4-regular planar graphs. In contrast, the enumeration of cubic planar graphs was completely solved in [5].
The contents of the paper are as follows. In Section 2 we find analogous results for planar maps, that is, connected planar graphs with a fixed embedding. They are simpler to derive and serve as a preparation for the results on planar graphs, while at the same time they are new and interesting by themselves. In Section 3 we find equations linking the generating functions of connected graphs, 2-graphs and 3-graphs; to this end we must consider multigraphs as well as simple graphs. In Section 4 we use singularity analysis in order to prove our main results on asymptotic enumeration and properties of random planar 2-graphs and 3-graphs. The analysis of the distribution of the degree of the root, which is technically more involved, is deferred to Section 5. We conclude with some remarks and open problems.
We assume familiarity with the basic results of analytic combinatorics as described in [9]. In particular, we need the following.
Transfer Theorem. If f (z) is analytic in a ∆-domain and satisfies, locally around its dominant singularity ρ, the estimate Quasi-powers Theorem. Let X n be a discrete parameter with associated bivariate generating function F (z, u), where u marks the parameter. Suppose that there is a representation in a bivariate ∆-domain. Let ρ(u) be the unique singularity of u → F (z, u), given by C(ρ(u), u) = 0. Then X n is asymptotically Gaussian with linear expectation and variance, and

Planar maps
We recall that a planar map is a connected planar multigraph embedded in the plane up to homeomorphism. A map is rooted if one of the edges is distinguished and given a direction. In this way a rooted map has a root edge and a root vertex (the tail of the root edge). We define the root face as the face to the right of the directed root edge. A rooted map has no automorphisms, in the sense that every vertex, edge and face is distinguishable. From now on all maps are planar and rooted. We stress the fact that maps may have loops and multiple edges. The enumeration of rooted planar maps was started by Tutte in his seminal paper [17]. Let m n be the number of rooted maps with n edges, with the convention that m 0 = 0. Then m n = 2 · 3 n (n + 2)(n + 1) The generating function M (z) = n≥0 m n z n is equal to Either from the explicit formula or from the expression for M (z) and the transfer theorem, it follows that m n ∼ 2 √ π n −5/2 12 n .
If m n,k is the number of maps with n edges and degree of the root face equal to k, then M (z, u) = m n,k u k z n satisfies the equation By duality, M (z, u) is also the generating function of maps in which u marks the degree of the root vertex. This is a convenient modifications of the usual equations for maps, where the empty map is also counted. The core C of a map M is obtained, as for graphs, by removing repeatedly vertices of degree one, so that C has minimum degree at leat two. Then M is obtained from C by placing a planar tree at each corner (pair of consecutive half-edges) of C. This is equivalent to replacing each edge with a non-empty planar tree rooted at an edge. The number t n of planar trees with n ≥ 1 edges is equal to the n-th Catalan number and the generating function T (z) = t n z n satisfies We define a 2-map as a map with minimum degree at least two, and a 3-map as a map with minimum degree at least three. Let h n and k n be, respectively, the number of 2-maps and 3-maps with n edges.
Theorem 2.1. The generating functions H(z) and K(z) of 2-maps and 3-maps, respectively, are given by The following estimates hold: where Proof. The decomposition of a map into its core and the collection of trees attached to the corners implies the following equation: The first summand corresponds to the case where the map is a tree, and the second one where the core is not empty: each edge is replaced with a non-empty tree whose root corresponds to the original edge. The factor is interpreted as follows. The first summand corresponds to the case where the root of the map is in the core, and the second one to the case where it is in a pendant rooted tree τ , which we place at the left-back corner of the root edge of the core. In this case there is a non-empty sequence of nonempty trees from the root edge e of τ to the root edge of the core, and the factor 2 distinguishes the two possible directions of e.
In order to invert the former relation let x = T (z), so that We obtain Let now C be a 2-map. The kernel K of C is defined as follows: replace every maximal path of vertices of degree two in C with a single edge (see Figure 2). Clearly K is a 3-map and C can be obtained by replacing edges in K with paths. It follows that The first term corresponds to the substitution of paths for edges, and the extra factor 1/(1 − z) indicates where to locate the new root edge in the path replacing the original root edge. The last term corresponds to cycles, whose kernel is empty. Inverting the relation x = z/(1 − z) we obtain In order to obtain asymptotic estimates for h n and k n we need to locate the dominant singularities of H(z) and K(z). The singularity of M (z) is at ρ = 1/12, and that of T (z) is at 1/4. Hence the singularity of H(z) is at σ = ρT (ρ) 2 = 5 − 2 √ 6. If follows that the singularity of K(z) . For future reference we display these basic constants, that is, the dominant singularities for 2-and 3-maps: The singular expansion of M (z) at the singularity z = 1/12 can be obtained directly from the explicit formula (3), and is equal to where Z = √ 1 − 12z. Plugging this expression into (8) and expanding gives where now X = 1 − x/σ. A similar computation using (10) gives The estimates for h n and k n follow directly by the transfer theorem and using the equality Γ(−3/2) = 4 √ π/3. Our next result is a limit law for the size of the core and the kernel in random maps.
Theorem 2.2. The size X n of the core of a random map with n edges, and the size Y n of the kernel of a random 2-map with n edges are asymptotically Gaussian with The size Z n of the kernel of a random map with n edges is also asymptotically Gaussian with Proof. If u marks the size of the core in maps then and immediate extension of (7) yields It follows that the singularity ξ(u) of the univariate function z → M (z, u) is given by We are in a situation where the quasi-powers theorem applies, so that the distribution is asymptotically Gaussian with linear expectation and variance. An easy calculation gives If now u marks the size of the kernel in 2-maps then an extension of (9) gives The singularity χ(u) of z → K(z, u) is now given by Again the quasi-powers theorem applies and we have The last statement concerning Z n follows by combining equations (11) and (12), obtaining an expression of M (z, u) in terms of K(z), and repeating the same computations as before for the corresponding singularity function.
It is interesting to compare the previous result with the known results on the largest block (2-connected components) in random maps [1]. The expected size of the largest block in random maps is asymptotically n/3, rather smaller than the size of the core. In other words, the core C consists of the largest largest block B together with smaller blocks attached to B comprising in total √ 6−1 3 n ≈ 0.4832n edges. An explanation for this is the presence of a linear number of loops, which belong to the core, but do not belong to the largest block.
Our next goal is to analyze the size of the trees attached to the core of a random map. Theorem 2.3. Let X n,k count trees with k edges attached to the core of a random map with n edges. Then X n,k is asymptotically normal and Proof. The generating function for trees, where variable w k marks trees with k edges, is equal to The scheme for the core decomposition is then It follows that the singularity ρ k (w k ) of the univariate function z → M (z, w k ) is given by the equation An easy calculation gives The first part of the proof is concluded by noticing that (1 − σ)/(σ(1 + σ)) = 4 + 5 3 √ 6. Finally, k≥1 α k = √ 6/3 follows from the closed form of the generating function for the Catalan numbers.
Recall that the size of the core is asymptotically 1−σ 1+σ n = √ 6 3 n. Hence the asymptotic probability that a random tree attached to the core has size k is It follows that if k ≫ log(n)/ log (3), the expected number α k n of trees of size k tends to zero. This indicates that the size L n of the largest tree attached to the core is at most log(n)/ log (3) with high probability. We are going to show that in fact L n / log(n) tends to log(3). Before that we need some preliminaries. In order to analyze the parameter L n , we use the theory of Boltzmann samplers as developed in [7]. We can model a random map (different from a tree) as follows: take a random 2-map and replace every edge with a rooted tree, in a way that the probability that the size of a tree is k equals β k , independently for each tree. For the sake of simplicity we only consider the trees placed in edges different from the root. In other words if m + 1 is the size of the core of a random map, then the sizes of the trees attached to the core are modeled as a sequence Y 1 , . . . , Y m of i.i.d. random variables, where P (Y i = k) = β k . Then the size of the largest tree attached to the core of a random map is equal to max{Y 1 , . . . , Y m }. In order to analyze this extremal parameter we follow the approach of Gao and Wormald for the analysis of the maximum vertex degree in random maps [2].
uniformly for |x| bounded with x + γ(m) an integer.
We need the following technical lemma from [2], based on the method of moments.
, are non-negative integer variables and there is a sequence of positive reals γ(m), and constants 0 < α, c < 1 such that (ii) for any fixed r and sequences Then there exists a function ω = ω(m) → ∞ (sufficiently slowly as m → ∞) so that the following holds. For k = ⌊γ − ω⌋, the total variation distance between the distribution of (X k , X k+1 , . . .), and that of ( We need to check that the random variables . , Y m are as in Theorem 2.4, satisfy the conditions of the lemma. (13), and let X 1 , . . . , X m be random variables such that X i counts occurrences of i in Y 1 . . . Y m . Then for any fixed r and sequences k i (m) such that |k i (m) − γ(m)| = O(1) the following equation holds fofr all i, j, and 0 otherwise. We have The sum q v(q) is bounded from below by [m] l1+···+lr , which are the tuples in which all the s are different. It is also bounded from above by [m] l1 · · · [m] lr , which is when all the k i (m) are equal. Both expressions are asymptotically equivalent, so that q v(q) ∼ m l1+···+lr , and In order to obtain the claimed result, first note that p k ∼ k −3/2 3 −k /( √ πσ). This can be easily checked using the Stirling's approximation. Since mγ −3 Proof of Theorem 2.4. The sequence X 1 , . . . , X m , where X i counts occurrences of i in Y 1 , . . . , Y m , satisfies the conditions of Lemma 2.5: condition (i) is easy to check, and condition (ii) follows from Lemma 2.6. Condition (iii) is proven as follows: Therefore, by Lemma 2.5 there exists a function ω = ω(m) → ∞ so that for k = ⌊γ − ω⌋, the total variation distance between the distribution of (X k , X k+1 , . . .), and that of ( is an integer. Then: And since k − γ(m) = x this concludes the proof.
Note that if we modify the constant C the result is the same, since the solution of the equation (1), regardless of the value of C. Hence if now m = cn we get the same expression. The size of the core of a random planar map is is between ((1 − σ)/(1 + σ) − ǫ)n and n for every ǫ > 0 with high probability. We can finally state our main result on this parameter. Theorem 2.7. Let L n be the size the largest tree attached to the core of a random map with n edges. Let γ(n) be such that uniformly for |x| bounded with x + γ(n) an integer.
Our last result in this section deals with the distribution of the degree of the root vertex in 2maps and 3-maps. We let M (z, u) be the GF of maps, where z marks edges and u marks the degree of the root vertex. Similarly, H(z, u) is the GF for 2-maps, and T (z, u) = 1/(1 − uz(T (z) + 1)) − 1 for trees, where again u marks the degree of the root. Then we have The first term corresponds to the case where the root belongs to the core: we replace each edge with a tree, and each edge incident to the root vertex is replaced with a possibly empty tree, where u marks the degree of the root. The term T (z) + 1 in the denominator ensures that an edge is not replaced twice with a tree. The factor T (z, u) + 1 allows to place a possibly empty tree in the root corner. The second term corresponds to the case where the root belongs to a tree attached to the core: the denominator 1 − T (z) encodes a sequence of trees going from the core to the root edge. The last term corresponds to the case where the core is empty, and therefore the map is a tree. If we change variables x = T (z) and w = u(T (u, z) + 1)/(T (z) + 1), the inverse is

The former equation becomes
The first terms are The relationship between H(z, u) and K(z, u) is simpler: Inverting gives and the first terms are K(z, u) = 2u 4 z 2 + (4u 3 + 5u 6 )z 3 + (9u 3 + 9u 4 + 15u 5 + 14u 8 )z 4 + · · · In order to analyze H(z, u) and K(z, u) we need the expansion of M (z, u) near the singularity ρ = 1/12. As we have seen, the expansion of M (z) near z = 1/12 is A simple computation by indeterminate coefficients gives The limiting probability that a random map has a root vertex (or face) of degree k is equal to Both coefficients can be estimated using transfer theorems and we get that the probability generating function of the distribution is given by Our goal is to obtain analogous results for 2-maps and 3-maps.
Theorem 2.8. Let p M (u) be as before, and let p H (u) and p K (u) be the probability generating functions for the distribution of the root degree in 2-maps and 3-maps, respectively. Then we have where σ = 5 − 2 √ 6, as in Theorem 2.1. Furthermore, the limiting probabilities that the degree of the root vertex is equal to k exist, both for 2-maps and 3-maps, and are asymptotically The correction terms uσ in p H (u) and u 2 σ in p K (u) are due to the fact, respectively, that 2-maps have no vertices of degree one and 3-maps no vertices of degree two.
Proof. Since M (z, u) satisfies (16) and H(x, w) satisfies (14), we obtain where Z = 1 − z/σ, and H 3 (u) can be computed as The probability generating function of the distribution is given by as claimed in the statement. Now by (15), K(u, z) satisfies where now Z = 1 − z/τ and K 3 (u) is The probability generating function of the distribution is given by The asymptotics of the distributions can be obtained from that of p M (u). The singularity of p M (u) is at u M = 6/5, and its expansion is computed from the explicit formula in (17) as where U = 1 − 5u/6 and P −3 = 1/(4 √ 10). The singularity of p H and p K is obtained by solving the equation giving u H = u K = 3/2. Hence, the exponential growth constants are w H = w K = 2/3. The singular expansion of p H (u) is obtained by composing (18) and (20), giving as a result where now U = 1 − u 2/3, and Q −3 = P −3 15(1 − σ)/8 = 3(1 − σ)/16. The singular expansion of p K (u) is obtained by composing (19) and (21) giving as a result where U is as before and The estimates for p H (k) and p M (k) follow directly by the transfer theorem.

Equations for 2-graphs and 3-graphs
In this section we find expressions for the generating functions of 2-and 3-graphs in terms of the generating function of connected graphs. The results are completely general and specialize to the generating functions of planar graphs, since a graph is planar if and on if its core its planar, and in turn the core is planar if and only if its kernel is planar. Let C(x, y) be the generating function of connected graphs, where x marks vertices and y marks edges. Denote by H(x, y) and K(x, y) the generating functions, respectively, of 2-graphs and 3-graphs. We will find equations of the form H(x, y) = C (A 1 (x, y), B 1 (x, y)) + E 1 (x, y) K(x, y) = C(A 2 (x, y), B 2 (x, y)) + E 2 (x, y), where A i , B i and E i are explicit functions.
From now on all graphs are labelled, and all generating functions are of the exponential type.

2-graphs.
Let G be a connected graph. The core C of G is obtained by removing repeatedly vertices of degree one, so that G is obtained from C by replacing each vertex of G with a rooted tree. The number T n of rooted trees with n edges is known to be n n−1 , and the generating function T (x) = T n x n /n! satisfies T (x) = xe T (x) .
The core of G can be empty, in which case G must be an (unrooted) tree. The number U n of unrooted trees is known to be n n−2 , and the generating function U (x) = u n x n /n! is equal to Theorem 3.1. Let h n be the number of 2-graphs with n vertices. Then H(x) = h n x n /n! is given by Proof. The decomposition of a graph into its core and the attached rooted trees implies the following equation: The first summand corresponds to the case where the core is non-empty, and the second summand corresponds to the case where the graph is a tree. In order to invert the former relation let x = T (z), so that We obtain Equation (23) can be extended by taking edges into account. The generating functions T (x, y) and U (x, y) are easily obtained as T (x, y) = T (xy)/y and U (x, y) = U (xy)/y, and a quick computation gives

3-graphs.
A multigraph is a graph where loops and multiple edges are allowed. As in the case of simple graphs, we define a k-multigraph as a connected multigraph in which the degree of each vertex is at least k. Let C be a 2-multigraph. The kernel K of C is defined as follows: replace every maximal path of vertices of degree two in C with a single edge. Clearly K is a 3-multigraph, and C can be obtained by replacing edges in K with paths. Let G be a multigraph. For each i ≥ 1, let α i be the number of vertices in G which are incident to exactly i loops, and let β i be the number of i-edges, that is, edges of multiplicity i. The weight of G is defined as This definition is justified by the fact that when replacing an i-edge with i different paths, the order of the paths is irrelevant. Similarly, when replacing a loop with a path, the orientation is irrelevant. Note that the weight satisfies 0 < w( G) ≤ 1, and moreover w( G) = 1 if and only if G is simple. With this definition, the sum K n of the weights of all 3-multigraphs with n vertices is finite.
As a preliminary step to computing the generating function of 3-graphs, we establish a relation between 3-multigraphs and connected multigraphs. In order to distinguish between edges of different multiplicity, we introduce infinitely many variables as follows. Let C n,m,l1,l2,... be the sum of the weights of connected multigraphs with n vertices, m loops and l i i-edges for each i ≥ 1. Define similarly K n,m,l1,l2,... for 3-multigraphs, and let C(x, z, y 1 , y 2 , . . .) = C n,m,l1,l2,... x n z m y l1 1 y l2 2 . . . /n! and K(x, z, y 1 , y 2 , . . .) = K n,m,l1,l2,... x n z m y l1 1 y l2 2 . . . /n!. The proof of Theorem 3.2 is quite technical and is given below. As a corollary we obtain the generating function of 3-graphs. Recall that C(x, y) is the generating function of connected graphs. Corollary 3.3. Let K n,m be the number of 3-graphs with n vertices and m edges. The generating function K(x, y) = K n,m x n y m /n! is given by where A(x, y) = xe (x 2 y 3 −2xy)/(2+2xy) , B(x, y) = (y + 1)e −xy 2 /(1+xy) − 1, and E(x, y) is as in Theorem 3.2.
Proof. Since the weight of a simple graph is one, the number of simple 3-graphs is equivalent to the number of weighted 3-multigraphs without loops or multiple edges. This observation leads to K(x, y) = K(x, 0, y, 0, . . . , 0, . . .).
Moreover, for each connected multigraph G, a connected simple graph G can be obtained by removing loops and replacing each multiple edge with a single edge. Then G is obtained from G by replacing each edge with a multiple edge, and attaching zero or more loops at each vertex. This can be encoded as where the exponential and the 1/i! terms take care of the weights. Finally, Equation (27) can be obtained by combining (28), (26) and (29).
We remark that a formula equivalent to (27) was obtained by Jackson and Reilly [12], using the principle of inclusion and exclusion. Our approach emphasizes the assignment of weights to multigraphs, which are needed in the various combinatorial decompositions.
Note that taking y = 1 in Equation (27) we obtain the univariate generating function K(x) of 3-graphs as where s = − xy 2 1 1 + xy 1 .
where s = xy 2 The first summand corresponds to the case where there is at least one vertex of degree ≥ 3, and thus the kernel is not empty. The other summands correspond to cycles (each vertex is of degree exactly two): from the logarithm encoding cycles we must take care of cycles of length one or two. If the kernel is not empty, we replace every edge and every loop with a path. The expression s encodes a nontrivial path, consisting of at least one vertex. Each loop can be replaced with either another loop, or a vertex and a double edge, or a path consisting of at least two vertices; these operations are encoded, respectively, by z, xy 2 and s. Note that if the kernel has an i-loop, then we can replace any of the loops with a path, in both orientations. Therefore there are 2i ways to obtain the same graph, which compensates the fact that the weight of the new graph will be 2i times the weight of the old graph. Each k-edge can be replaced with a j-edge and k − j nontrivial paths, where 0 ≤ j ≤ k. There are (k − j)! ways to obtain the same graph, and the weight becomes k!/j! times the previous weight. Therefore y k is replaced with k j y j s k−j , for j = 0, . . . , k. A simple computation shows that inverting (32) gives (31), as claimed.
As mentioned before, Theorem 3.1 and Corollary 3.3 hold for planar graphs as well. In the next section we use them to enumerate and analyze planar 2-and 3-graphs.

Planar graphs
In this section we follow the ideas of Section 2 on planar maps in order to obtain related results for planar 2-graphs and 3-graphs. The asymptotic enumeration of planar graphs was solved in [10]. From now on we assume that we know the generating function C(x, y) of connected planar graphs, where x marks vertices and y marks edges, as well as its main properties, such as the dominant singularities and the singular expansions around them.
In this section we use the equations obtained in Section 3 to compute some parameters in planar graphs. Most of the computations will be analogous to the ones of maps, but technically more involved. In order to compare the following results, we recall [10] that the number of connected planar graphs is c n ∼ κn −7/2 γ n , where κ ≈ 0.4104 · 10 −5 and γ ≈ 27.2269. As expected, there will be slightly fewer connected 2-graphs and 3-graphs than connected planar graphs. Besides, the expected degree of 2-graphs and 3-graphs will be slightly higher.

Planar 2-graphs
We start our analysis with planar 2-graphs. The analysis for 3-graphs in the next subsection is a bit more involved.
Proof. Recall Equation (23) from Section 3: In order to obtain an asymptotic estimate for h n we need to locate the dominant singularity of H(x). The singularity of C(x) is ρ = γ −1 ≈ 0.0367 [10]. Hence the singularity of H(x) is at σ = T (ρ) ≈ 0.0382. Therefore, the exponential growth constant of h n is γ 2 = σ −1 ≈ 26.2076. Note that we use the same symbol σ as in Section 2 for maps, but they correspond to different constants. No confusion should arise and it helps emphasizing the parallelism between planar maps and graphs. The singular expansion of C(x) at the singularity x = ρ is where X = 1 − x/ρ, and C 5 ≈ −0.3880 · 10 −5 is computed in [10]. Plugging this expression into (23) and expanding gives where now X = 1 − x/σ and H 5 = C 5 (1 − σ) 5/2 ≈ −0.3520 · 10 −5 . The estimate for h n follows directly by the transfer theorem.
Our next result is a limit law for the number of edges in a random planar 2-graph. We recall [10] that the expected number of edges in random connected planar graphs is asymptotically µn, where µ ≈ 2.2133, and the variance is λn with λ ≈ 0.4303.
Theorem 4.2. The number X n of edges in a random planar 2-graph with n vertices is asymptotically Gaussian with E X n ∼ µ 2 n ≈ 2.2614n, Var X n ∼ λ 2 n ≈ 0.3843n.

Proof. Equation (25) from Section 3
H(x, y) = C(xe −xy , y) − x + x 2 y 2 implies that the singularity σ(y) of the univariate function x → H(x, y) is given by where ρ(y) is the singularity of the univariate function x → C(x, y). An easy calculation gives which provides the constant for the expectation. Similarly This value can be computed from the known values of µ, λ and σ.
Next we determine a limit law for the size of the core and the kernel in random connected planar graphs.
The size X n of the core of a random connected planar graph with n edges is asymptotically Gaussian with Proof. The generating function C(x, u) of connected planar graphs, where u marks the size of the core, is given by It follows that the singularity ξ(u) of the univariate function x → C(x, u) is given by the equation We can isolate ξ(u) obtaining the explicit formula An easy calculation gives Our next goal is to analyze the size of the trees attached to the core of a random connected planar graph.
Theorem 4.4. Let X n,k count trees with k vertices attached to the core of a random connected planar graph with n vertices. Then X n,k is asymptotically normal and and β k is described in the proof.
Proof. The generating function of trees where variable w k marks trees with k vertices is equal to where T k = k k−1 /k! is the k-th coefficient of T (x). The composition scheme for the core decomposition is then C(x, w k ) = H(T (x, w k )) + U (x).
It follows that the singularity ρ k (w k ) of the univariate function x → C(x, w k ) is given by the equation An easy calculation gives As expected, k≥0 α k = 1 − σ, since there are σn vertices not in the core, and therefore there are (1 − σ)n trees attached to the core. Moreover, k≥0 kα k = 1, since a connected graph is the union of the trees attached to its core.
To conclude this section, we consider as for maps the parameter L n equal to the size of largest tree attached to the core of a random planar connected graph. The estimate for the α k is We can perform exactly the same analysis as in Section 2 in order to obtain a law for L n .
Theorem 4.5. Let L n be the size of largest tree attached to the core of a random planar connected graph. Let γ(n) be such that uniformly for |x| bounded with x + γ(n) an integer.

Planar 3-graphs
We recall again that the generating function of connected planar graphs C(x, y), where x marks vertices and y marks edges, was computed in [10].
Proof. Recall Equation (30) from Section 3: where In order to obtain an estimate for k n we need to locate the dominant singularity of K(x). The singularity curve of C(x, y) is given by (X(t), Y (t)), where t ∈ (0, 1) and X, Y are explicit functions defined in [10]. Hence the singularity τ of K(x) is obtained by solving the equations The solution τ of the equation can be computed numerically and is τ ≈ 0.0469. The exponential growth constant is then γ 3 = τ −1 ≈ 21.3102.
Our next result is a limit law for the number of edges in a random planar 3-graph.
Theorem 4.7. The number X n of edges in a random planar 3-graph with n vertices is asymptotically Gaussian with Proof. Recall Equation (27) from Section 3: K(x, y) = C (A(x, y), B(x, y)) + E(x, y), where A(x, y) = xe (x 2 y 3 −2xy)/(2+2xy) , B(x, y) = (y + 1)e −xy 2 /(1+xy) − 1, It follows that the singularity τ (y) of the univariate function x → K(x, y) is given by the equation where ρ(y) is as before the singularity of x → C(x, y). The value of τ (1) = τ is already known. In order to compute τ ′ (1) we differentiate and obtain Solving for τ ′ (1) we obtain Since ρ = X • Y −1 , where X and Y are explicit functions defined in [10], ρ ′ (y) can be computed as X ′ (Y −1 (y))/Y ′ (Y −1 (y)). After some calculations we finally get a value of τ ′ (1) ≈ −0.1129 and Using the same procedure we can isolate τ ′′ (1) ≈ 0.3700 and obtain the variance as Next we determine the limit law for the size of the kernel in random planar 2-graphs.
Theorem 4.8. The size Y n of the kernel of a random planar 2-graph with n edges is asymptotically Gaussian with Proof. Recall that the decomposition of a simple 2-graph into its kernel gives If u marks the size of the kernel then Composing with Equations (26) and (29) we get and F (x, u) is a correction term which does not affect the singular analysis. It follows that the singularity χ(u) of the univariate function x → H(x, u) is given by the equation If we differentiate the former expression and replace u with 1 we get A x (σ, 1)χ ′ (1) + A y (σ, 1) = ρ ′ (1)(B x (σ, 1)χ ′ (1) + B y (σ, 1)).

Degree distribution
In this section we compute the limit probability that a vertex of a planar 2-graph or 3-graph has a given degree. In order to do that, we compute the probability distribution of the root of a rooted planar 2-graph and 3-graph. Since every vertex is equally likely to be the root, we conclude that the average distribution is the same. Note that this is not true for maps, so in this section we only compute the distribution for graphs. This section is rather technical, especially the part of 3-graphs, so that is why we separate its content from that of Section 4.
Let c • n be the number of rooted connected planar graphs with n vertices, i.e., c • n = n · c n . Let C • (x) = c • n x n = xC ′ (x) be its associated generating function. Let c • n,k be the number of rooted connected planar graphs with n vertices and such that the root degree is exactly k. Let C • (x, w) = c • n,m x n u m be its associated generating function. The limit probability d k that the root vertex has degree k can be obtained as Therefore, the probability distribution p(w) = d k w k can be obtained from the knowledge of C • (w, u). In [6] this function is computed, and d k is proven to be asimptotically where c ≈ 3.0175 and q ≈ 0.6735 are computable constants. Our goal is to obtain similar results for 2-graphs and 3-graphs, by respectively computing generating function H • (x, w) and K • (x, w) in terms of C • (x, w).

2-graphs
Theorem 5.1. Let h • n,k be the number of rooted 2-graphs with n vertices and with root degree k. Let H • (x, w) = h • n,k x n w k be its associated generating function. The following equation holds Proof. The decomposition of a graph into ins core and the attached rooted trees implies the following equation: where T (z, w) = z · e wT (z) is the generating function of rooted trees where w marks the degree of the root. The first addend corresponds to the case where the root is in the core. In this case, the degree of the graph root is the degree of the core root plus the degree of the root of its appended tree. The second addend corresponds to the case where the root is in an attached tree. In this case there is a sequence of trees between the core and the root, and finally a rooted tree. The degree of the graph root is the degree of the root of the rooted tree plus one. The last addend corresponds to the case where the graph is a tree, and therefore its core is empty. In order to invert the former relation let x = T (z) so that After some calculations we obtain The probability distribution p(w) can be computed using transfer theorems. The expansion of C • (x, w) near the singularity x = ρ gives the following equation where X = 1 − x/ρ. The probability distribution can be computed as Our goal is to obtain the same result by applying the relation obtained in (38).
Theorem 5.2. Let e k be the limit probability that a random vertex has degree k in a 2-graph.
Let p H (w) = e k w k be its probability distribution. Let p(x) be as before. The following equation holds: where σ = T (ρ), as in Theorem 4.1. Furthermore, the limiting probability that the degree of a random vertex is equal to k exists, and is asymptotically where q ≈ 0.6735 and ν 2 ≈ 3.0797.
Proof. Since C • (x, w) satisfies (39), and H • (x, w) satisfies (38), we obtain where X = 1 − x/σ, and H 3 (w) is computed as The probability generating function of the distribution is given by The asymptotics of the distribution can be obtained from p(w). The singularity of p(w) is obtained in [6] as r ≈ 1.4849. The expansion of p(w) near the singularity is computed as where P −1 ≈ 5.3484 is a computable constant, and W = 1 − w/r. Plugging this expression into (40) we get where Q −1 = P −1 e σ(1−r) /(1 − σ) ≈ 5.4586. The estimate for p H (k) follows directly by singularity analysis.

3-graphs
In order to prove a similar result for 3-graphs, we need to extend the generating function C • (x.w) so that it takes edges into account. This function C • (x, y, w) was computed in [6], and our goal is to obtain the analogous generating function for 3-graphs, K • (x, w), in terms of We remark that the expression given in [6] for C • (x, y, w) is extremely involved and needs several pages to write it down.
Theorem 5.3. Let k • n,k be the number of rooted 3-graphs with n vertices and with root degree k. Let K • (x, w) = k • n,k x n w k be its associated generating function. The following equation holds where and A 0 (x), A 1 (x), A 2 (x) are analytic functions.
In order to prove this theorem we need some technical lemmas that relate different classes of graphs.
Lemma 5.4. Let C • (x, w, z, y 1 , . . . , y k , . . .) be the generating function of rooted connected planar weighted multigraphs where x marks vertices, w marks the root degree, z marks loops, and y k marks k-edges. The following equation holds Proof. Given a simple connected planar graph G, a connected planar multigraph can be obtained from G by replacing each edge with a multiple edge, and placing 0 or more loops in each vertex (see proof of Corollary 3.3 for details). In the case of rooted graphs, if we replace an edge incident to the root with a i edge, its root degree is increased in i − 1. Therefore, instead of replacing such an edge with a multiple edge with generating function y i /i!, we replace it with a multiple edge with generating function w i y i /i!. Similarly, when we add a loop incident to the root vertex, the root degree is increased by 2. Therefore, its associated generating function is not z, but zw 2 .
Proof. The decomposition of a planar connected weighted multigraph into its core and the attached rooted trees implies the following equation: C • (x, w, z, y 1 , . . . , y k , . . .) = H • (T (x, y 1 ), w, z, y 1 , . . . , y k , . . .) T (x, y 1 , w) T (x, y 1 ) + + H • (T (x, y 1 ), z, y 1 , . . . , y k , . . .) wT (x, y 1 , w) 1 − T (x, y 1 ) + T (x, y 1 , w), where T (x, y) = T (xy)/y is the generating function of rooted trees where x marks vertices and y marks edges, and T (x, y, w) = T (xy, w)/y is the generating function of rooted trees where x marks vertices, y marks edges, and w marks the root degree. The justification of this relation is analogous to the proof of Theorem 5.1, as well as the inverse.
for a given function A(x), and where s = −x/(1 + x).
Since we know that a 3-graph has no vertices of degree 0, 1 or 2, we can choose suitable values of a 0 , a 1 and a 2 such that the probability distribution p K (w) = f k w k satisfies f 0 = f 1 = f 2 = 0. The function C 3 (y, w) is described in [6], and every other function that appears in the previous expression is explicit. Therefore, p K is computable, as we wanted to prove.
We remark that p K (w) is expressed in terms of C 3 (x, w), which is a very involved (although elementary) function, given in the appendix in [6].

Concluding remarks
Most of the results we have obtained can be extended to other classes of graphs. Let G be a class of graphs closed under taking minors such that the excluded minors of G are 2-connected. Interesting examples are the classes of series-parallel and outerplanar graphs. Given such a class G, a connected graph is in G if and only if its core is in G. Hence Equation (23) also holds for graphs in G. Using the results from [4], we have performed the corresponding computations for the classes of series-parallel and outerplanar graphs (there are no results for kernels since outerplanar and series-parallel have always minimum degree at most two). The results are displayed in the next table, together with the data for planar graphs. The expected number of edges is µn, and the expected size of the core is κn. It is worth remarking that the size of the core is always linear, whereas the size of the largest block in series-parallel and outerplanar graphs is only O(log n) [11,16].

Graphs
Growth constant µ (edges) κ (core) The k-core of a graph G is the maximum subgraph of G in which all vertices have degree at least k. Equivalently, it is the subgraph of G formed by deleting repeatedly (in any order) all vertices of degree less than k. In this terminology, what we have called the core of a graph is the 2-core. Using the results from [10] it is not difficult to show that the 3-core, 4-core and 5-core of a random planar graph have all linear size with high probability (there is no 6-core since a planar graph has always a vertex of degree at most five). The interesting question is however whether the k-core has a connected component of linear size (as is the case for k = 2). We have performed computational experiments on random planar graphs, using the algorithm described in [8], and based on the results we formulate the following conjecture.
Conjecture. With high probability the 3-core of a random planar graph has one component of linear size. With high probability the components of the 4-core of a random planar graph are all sublinear.
We have not been able to prove neither of the conjectures. As opposed to the kernel, the 3-core is obtained by repeatedly removing vertices of degree two. These deletions may have long-range effects that appear difficult to analyze. Even more challenging appears the analysis of the 4-core.