Extremal Statistics on Non-Crossing Configurations

Transcription

1 Extremal Statistics on Non-Crossing Configurations Michael Drmota Anna de Mier Marc Noy Abstract We analye extremal statistics in non-crossing configurations on the n vertices of a convex polygon. We prove that the maximum degree and the largest component are of logarithmic order, and that, suitably scaled, they converge to a well-defined constant. We also prove that the diameter is of order n. The proofs are based on singularity analysis, an application of the first and second moment method, and on the analysis of iterated functions. 1 Introduction and preliminaries Let p 1,..., p n be the vertices of a convex polygon in the plane, labelled counterclockwise. A noncrossing graph (or configuration) is a graph on these vertices such that when the edges are drawn as straight lines the only intersections occur at vertices. The root of a graph is vertex p 1. We call the edge p 1 p n (if present) the root edge. From now on, all graphs are assumed to be non-crossing graphs. A triangulation is a graph with the maximum number of edges and it is characteried by the fact that all internal faces are triangles. A dissection is a graph containing all the boundary edges p 1 p, p p 3,..., p n p 1 ; a single edge p 1 p is also considered a dissection (see Figure 1). From a graph theoretic point of view, dissections are the same as -connected graphs. The root region of a dissection is the internal region adjacent to the root edge. Figure 1: From left to right: a triangulation, a -connected graph (dissection), a connected graph, and an arbitrary graph. The enumerative theory of non-crossing configurations is an old subject, going back to Euler; see, for instance, Comtet s book [4] for an account of classical results. Flajolet and Noy [11] reexamined these problems using the tools from analytic combinatorics [13] in a unified way. They showed that for all natural classes under consideration the number of non-crossing graphs with n vertices is asymptotically of the form cn 3/ γ n, for some positive constants c and γ. In addition, many basic parameters obey a Gaussian limit law with linear expectation and variance. These include: number of edges, number of components, Research partially supported by Projects MTM and DGR009-SGR1040. Technische Universität Wien, Vienna. michael.drmota@tuwien.ac.at Universitat Politècnica de Catalunya, Barcelona. anna.de.mier@upc.edu Universitat Politècnica de Catalunya, Barcelona. marc.noy@upc.edu 1

2 number of leaves in trees and number of blocks in partitions. The proofs in [11] are based on perturbation of singularities and extensions of the Central Limit Theorem. In this paper we take a step further and analye more complex parameters, specially extremal parameters. Some results have been obtained previously for triangulations [6, 14] and trees [5, 16], but here we aim at a systematic treatment of the subject, covering the most important extremal parameters and proving limit laws whenever possible. Our main results are summaried as follows. For graphs, connected graphs and -connected graphs, the degree of the root vertex converges to a discrete law. More precisely, if p k is the probability that the root has degree k, then pk w k = A(w) (1 qw), where A(w) is a polynomial of degree two and q is a quadratic irrational with 0 < q < 1. It follows that the tail of the distribution satisfies, for a suitable constant c > 0, p k ckq k, as k. For graphs, connected graphs and -connected graphs, the maximum degree n is of logarithmic order. More precisely, for each class under consideration there exists a quadratic irrational q > 0 such that n log n 1 log(q 1 ) in probability. The largest connected component M n in graphs is of logarithmic order: there exists a quadratic irrational q > 0 such that M n log n 1 log(q 1 ) in probability. For triangulations, connected graphs and -connected graphs, the diameter D n is of order n. For each class under consideration, there exist constants 0 < c 1 < c such that c 1 n < EDn < c n. These results reflect the tree-like nature of non-crossing configurations. In particular, the diameter is of order n, like the height of plane trees. The expected maximum degree in triangulations was shown to be asymptotically log n/ log in [6] (much more precise results were obtained in [14]). To our knowledge, the diameter of random configurations has not been studied before, even in the basic case of triangulations. In the rest of this section we collect several technical preliminaries needed in the paper. In Section we analye the degree of the root vertex. Sections 3 and 4 are devoted to the maximum degree and the sie of the largest component, respectively, and are based on the first and second moment method. Finally, in Section 5 we analye the diameter, making use of iterated functions. 1.1 Generating functions We denote by G() and C() the generating functions for arbitrary and connected graphs, respectively, counted by the number of vertices. Furthermore, let B() be the generating function for -connected graphs, where marks the number of vertices minus one. We have the following relations for the generating functions. The first one reflects the decomposition of a dissection as a sequence of dissections attached to the root region, as in [11]. See Figure 1.1 (left) for an illustration. B() = + B() 1 B(). (1)

3 Of the two solutions of this equation, only one is a power series, hence B() = which has a square-root singularity at = 3. The next equation encodes the decomposition of a connected graph into -connected components, also called blocks; see Figure 1.1 (center). C() = () 1 B(C() /). (3) Indeed, a connected graph consists of a root and a sequence of blocks containing the root, in which each vertex is substituted by a pair of connected graphs (to the left and to the right) with one vertex identified. Eliminating B we obtain C() 3 + C() 3C() + = 0, (4) in accordance with [11]. The function C() has a dominant singularity at = 3/18, also of square-root type. The decomposition of an arbitrary graph into connected components gives, as Figure : Left: a dissection consisting of a root face (in gray) with an arbitrary dissection glued at each non-root edge. Center: the root of the connected graph belongs to three blocks (in gray); to any other vertex of these blocks there are two connected graphs attached, possibly reduced to a single vertex. Right: between every two consecutive vertices of the root component (in gray) there is an arbitrary graph, possibly empty. In all three cases the root is the vertex on top. in [11], the equation See Figure 1.1 (right) for an example. It follows that G() satisfies which leads to G() = 1 + C(G()). (5) G + ( 3 )G = 0, G() = (6) This function is singular at = 3/. Let us remark that G() = B(). We summarie the above discussion in the following table: Function Equation Singularity G() G + ( 3 )G = 0 C() C 3 + C 3C + = B() B (1 + )B + = 0 3 3

4 1. Singularity analysis In what follows we will make use of power series of the square-root type. They are power series y() with a square-root singularity at 0 > 0, that is, y() admits a local representation of the form y() = g() h() 1 / 0, (7) for 0 < ε for some ε > 0, and arg( 0 ) > 0, where g() and h() are analytic and non-ero at 0. Moreover, y() can be analytically continued to the region D( 0, ε) = { C : < 0 + ε} \ [ 0, ). (8) Let [ n ]y() be the n-th coefficient in y(). The Transfer Theorem of Flajolet and Odlyko (see [13]), implies the following estimate: [ n ]y() h( 0) π n 3/ n 0. For the analysis of the root degree we need a particular application of this estimate, whose proof can be be found in [9]. Lemma 1.1. Let f() = n 0 a n n denote the generating function of a sequence a n of nonnegative real numbers and assume that f() has exactly one dominating square-root singularity at = ρ of the form f() = g() h() 1 /ρ, where g() and h() are analytic at = ρ and f() has an analytic continuation to the region D( 0, ε) for some ε > 0. Further, let H(, t) denote a function that is analytic for < ρ + ε and t < f(ρ) + ε, and such that H t (ρ, f(ρ)) 0, where H t stands for the derivative with respect to t. Then the function f H () = H(, f()) has a power series expansion f H () = n 0 b n n and the coefficients b n satisfy b n lim = H t (ρ, f(ρ)). (9) n a n In our applications, H(, t) depends also on an additional variable w, that can be treated as a parameter. 1.3 First and second moment method In order to obtain results for the maximum statistics of the root degree we follow the methods of [10]. They are based on the so-called first and second moment method [1]. Lemma 1.. Let X be a discrete random variable on non-negative integers with finite first moment. Then P{X > 0} min{1, E X}. Furthermore, if X is a non-negative random variable which is not identically ero and has finite second moment then (E X) P{X > 0} E (X ). We apply this principle for the random variable Y n,k that counts the number of vertices of degree greater than k in a random graph with n vertices. This variable is closely related to the maximum degree n by Y n,k > 0 n > k. 4

5 One of our aims is to get bounds for the expected maximum degree E n. Due to the relation E n = k 0 P{ n > k} = k 0 P{Y n,k > 0} we are led to estimate the probabilities P{Y n,k > 0}, which can be done via the first and second moment methods by estimating the first two moments E Y n,k and E Y n,k. Actually, we work with the probabilities d n,k that a random vertex in a graph of sie n has degree k. They are related to the first moment by E Y n,k = n l>k d n,l. (10) Similarly we deal with probabilities d n,k,l that two different randomly selected vertices have degrees k and l. Here we have E Yn,k = n d n,l + n(n 1) d n,l1,l. l>k l 1,l >k The following technical lemma subsumes some results from [10] and it is stated according to our needs in this paper. The full proof can be found in Theorem 1.1 and Lemmas 3.1 and 3. from [10]. The proof of the version below is a slight adaptation. It guarantees that, if the generating functions associated to the degree of the root and the degree of a secondary vertex have a certain local expansion of square-root type, then automatically the maximum degree is c log n for a well-defined constant c. For our functions these conditions are not difficult to check. Lemma 1.3. Let f(, w) = n,k f n,k n w k be a double generating function and g(, w, t) = n,k,l g n,k,l n w k t l be a triple generating function of non-negative numbers f n,k and g n,k,l such that the probabilities d n,k and d n,k,l that a random vertex in a graph of sie n has degree k, and that two different randomly selected vertices have degrees k and l, respectively, are given by d n,k = f n,k f n and d n,k,l = g n,k,l g n, where f n = k f n,k and g n = k,l g n,k,l. Suppose that f(, w) can be represented as f(, w) = G(, Z, w) 1 y()w, (11) where Z = 1 / 0, y() is a power series with non-negative coefficients of square-root type, y() = g() h() 1 / 0, where 0 < g( 0 ) < 1 and the function G(, v, w) is analytic in the region D = {(, v, w) C 3 : < 0 + η, v < η, w < 1/g( 0 ) + η} for some η > 0 and satisfies G( 0, 0, 1/g( 0 )) 0 and G( 0, 0, 1)h( 0 ) G v ( 0, 0, 1)(1 g( 0 )). Furthermore suppose that g(, w, t) can be represented as g(, w, t) = H(, Z, w, t) Z (1 y()w) (1 y()t), (1) where H(, v, w, t) is non-ero and analytic at (, 0, w, t) = ( 0, 0, 1/g( 0 ), 1/g( 0 )) and (, 0, w, t) = ( 0, 0, 1, 1), and that ( ) h( 0 )G( 0, 0, 1/g( 0 )) = H( 0, 0, 1/g( 0 ), 1/g( 0 )). (13) g( 0 )(G( 0, 0, 1)h( 0 ) G v ( 0, 0, 1)(1 g( 0 ))) H( 0, 0, 1, 1) 5

6 Let n denote the maximum degree of a random graph in this class of sie n. Then we have n log n 1 log g( 0 ) 1 in probability (14) E n 1 log n (n ). (15) log g( 0 ) 1 We do not go into the details of the proof. We just mention that the main intermediate step is to prove that d n,k ckg( 0 ) k and d n,k,l d n,k d n,l c klg( 0 ) k+l, uniformly for k C log n and l C log n (for a certain constant c > 0 and an arbitrary constant C > 0). With the help of these asymptotic relations one gets asymptotic expansions for E Y n,k and E Y n,k that can be used to estimate the probabilities P{ n > k} from below and above and which lead to the final result. We direct the reader to [10] for a detailed discussion of the lemma and its proof, based on the estimation of Cauchy integrals. The key conditions in the lemma are equations (11) and (1) for the shape of the generating functions marking the degree of one and two vertices, respectively, and the square-root type of the univariate function y(). In (11) we find a linear factor in w in the denominator, and in (1) we find a quadratic factor both in w and in t, and a factor 1 / 0. In Section 3 we point out how these conditions are satisfied in our case. 1.4 Iterated functions The results on the diameter are based on the following lemma on iterated functions. Such a lemma was first studied in the analysis of the height of random trees, as in [1]. A basic example is given by the class T of plane trees (rooted trees in which the children of a node are ordered from left to right). Let T [k] the class of plane trees with height at most k. In the terminology of the symbolic method [13], we have the decomposition T [k+1] = {ρ} Seq(T [k] ), where Seq denotes the sequence construction and ρ represent the root of the tree. This is because the subtrees attached to the children of the root of a tree in T [k+1] have height at most k. This translates into an equation for the associated generating functions: T [k+1] () = 1 T [k] (). This is precisely the kind of equations that are covered in the lemma. Notice also that the generating function of trees with height exactly k is equal to T [k] T [k 1]. In the next statement, the notation A() B() means that the coefficients of A() = n a n n and B() = n b n n satisfy a n b n. Lemma 1.4. Suppose that F (, t) is an analytic function at (, t) = (0, 0) such that the equation T () = F (, T ()) (16) has a solution T () that is analytic at = 0 and has non-negative Taylor coefficients. Suppose that T () has a square-root singularity at = 0 and can be continued to a region of the form (8), such that F t ( 0, t 0 ) = 1, F ( 0, t 0 ) 0, and F tt ( 0, t 0 ) 0, where t 0 = T ( 0 ). Let T 0 () be a power series with 0 T 0 () T () such that T 0 () is analytic at = 0, and let T k (), k 1 be iteratively defined by Assume that T k 1 () T k () T (). T k () = F (, T k 1 ()). (17) 6

7 Let H n be an integer valued random variable that is defined by P{H n k} = [n ] T k () [ n ] T () for those n with [ n ] T () > 0. Then π E H n 0 F ( 0, t 0 )F tt ( 0, t 0 ) n1/. In the previous statement, H n is a kind of generalied height parameter. Recursion (17) is analogous to our previous equation for trees of height at most k. The quotient [ n ]T k ()/[ n ]T () is the proportion of elements of sie n and height at most k among the total number of elements of sie n. The series T () is in a sense a limit of the T k (), and is defined by the fixed point equation (16). The lemma is a direct extension of the results in [1]; see also Theorems 4.8 and 4.59 in [7] for the proof techniques. The previous lemma can be more precise, in the sense that H n / n converges to the maximum of a Brownian excursion of duration one (a Brownian motion conditioned to be positive and to take the value 0 at time 1). However, the estimate on the expectation is sufficient for our needs. Degree of the root In this section we determine the asymptotic distribution of the degree of the root vertex in graphs, and also in connected and -connected graphs. The probability generating function is in all cases a rational function with a quadratic factor in the denominator. This implies that the probability that the root has degree k is asymptotically, for large k, of the form ckq k, where q < 1 is a constant that depends on the class of graphs under consideration. Let B(, w), C(, w) and G(, w) be the corresponding generating functions, where w counts the degree of the root vertex p 1. A simple adaptation of the basic equations for B(), C() and G() gives wb(, w)b() B(, w) = w +, 1 B() C(, w) = 1 B(C() /, w), (18) G(, w) = 1 + C(G(), w). (19) In the first equation, the term B(, w) on the right singles out the only dissection that contributes to the degree of the root vertex. In the second equation, the sum of the degrees of the blocks containing the root is added. And in the last equation, only the component containing the root contributes to its degree. In particular, we have Let d B k is, B(, w) = w(1 B()) 1 (1 + w)b() = w 1 w ( ). (0) be the limiting probability that the root vertex in a -connected graph has degree k, that d B k = lim n [ n ][w k ]B(, w) [ n, ]B() and define d C k and dg k analogously. Next we find the probability generating functions for the d B k, dc k indeed exist. The following result can be found also in [9] and []. and dg k, and show that they 7

8 Theorem.1. The limiting distribution of the root degree in -connected graphs is given by p B (w) = k 1 d B k w k = (3 )w (1 ( 1)w). Proof. We use Lemma 1.1 with f() = B() and H(, t) = w(1 t)/(1 t wt). We know that B() has a unique dominating square-root singularity at ρ = 3. By substituting into Equation () we find that B(ρ) = ( )/. The denominator of H(, t) does not vanish for t < B(ρ) + ε and w in a neighbourhood of 1, hence the conditions of Lemma 1.1 hold. The result follows then by differentiating H(, t) with respect to t and evaluating at (ρ, B(ρ)). Notice that p B (1) = 1, so that it is indeed a probability distribution. Now we prove a similar result for connected and arbitrary graphs. Theorem.. The limiting distributions of the root degree in connected and arbitrary graphs are given, respectively, by p C (w) = k 1 p G (w) = k 1 d C k w k = (1 1 3 ) d G k w k = ( 1) Proof. We use a slight adaption of Lemma 1.1 applied to H(, t) = 1 B(t /, w) w(w ) (1 (1 1 3 )w), (1 + w) (1 ( 1)w). and f() = C(). Actually in this case the assumption that H(, t) is analytic for < ρ + ε and t < f(ρ) + ε is not satisfied. However, it is sufficient to observe that B(, w) is analytic for < B(ρ) /ρ + ε (note also that B() /) B(ρ) /ρ for ρ). This is easy to observe since B(ρ) /ρ and the square root in B() (or B(, w)) is analytic for < Consequently the dominant singularity of C() is ρ = 3/18 and it is of the square-root type. We need to find the derivative H t (, t) and evaluate it at the point (ρ, C(ρ)). Substituting = ρ into Equation (4) and noting that the value of C(ρ) must be positive we conclude that C(ρ) = ( 3 1)/6. We next find the equation satisfied by H(, t) eliminating from Equations (0) and (). ( + t w t w w ) H(, t) + ( w + t 4 w 3 t w ) H(, t) 3 w t w + t w = 0. By differentiating with respect to t we find a linear equation for H t (, t) which leads to the expression in the statement of the theorem upon substituting t = C(ρ), = ρ and H = C(ρ, w). To find this last value, we simply substitute = ρ in the equation above and of the two solutions we pick the one that evaluated at w = 1 gives C(ρ), which is ( 3 1)(w 3)/(6( w 3 3)). The result for p G (w) can be proved in a similar way, but we prefer a more combinatorial proof, that explains in addition why we find the same value 1 in the denominator of both p G (w) and p B (w). As observed in [11], an arbitrary graph can be obtained from a polygon dissection by removing a subset of the edges of the polygon. If we put b n = [ n 1 ]B() and g n = [ n ]G(), we have that g n = b n n. Let b n,k = [ n 1 ][w k ]B(, w) and g n,k = [ n ][w k ]G(, w). We have the following relationship g n,k = b n,k n + b n,k+1 n + b n,k+ n, 8

9 which reflects the different possibilities for obtaining a graph with root degree equal to k from a dissection of root degree k, k + 1 or k +. By dividing both sides of this equation by g n = b n n we obtain 4 g n,k g n = b n,k b n + b n,k+1 b n + b n,k+ b n. This allows us in particular to express the limiting probability distribution for arbitrary graphs in terms of that for -connected graphs, giving the expression in the statement. Notice again that p C (1) = 1 and p G (1) = 1. In the next table we give approximate values for the different probabilities involved. k connected connected arbitrary Table 1: Probability that the degree of the root is k in -connected, connected and arbitrary graphs. From the previous explicit expressions it is immediate to obtain the tail of the distribution. In all cases it is of the form ckq k, for suitable c and q. Corollary.3. We have the following estimates, as k : 3 Maximum Degree p B k k( 1) k ( ) ) k p C k k (1 3 1 p G k k( 1) k In this section we show that the maximum degree is of order log n and, suitably scaled, converges to a well-defined constant. Theorem 3.1. The maximum degree n for -connected, connected and arbitrary non-crossing graphs satisfies n log n c in probability, where c = 1/ log(q 1 ) and q = 1 for -connected and arbitrary graphs, and q = 1 1/ 3 for connected graphs. In all cases we also have E n c log n as n. Proof. We must show that the associated generating functions satisfy the conditions imposed on f(, w) and on g(, w, t) in Lemma 1.3. We treat first the case of a single root, and then that of a root plus a secondary vertex. We rewrite (0) as w B(, w) = 1 y B ()w, where y B () = ( )/. Now we consider C(, w). From the first equality in (0), the expression for C(, w) in (18) becomes (1 (1 + w)b(c /) C(, w) = 1 (1 + w + wc /)B(C /) wc /, 9

10 where we have set C = C() for brevity. Eliminating from equation (3), we arrive at C(, w) = + w wc() 1 y C ()w, where y C () = (C() + C() )/. For G(, w), from (19), the expression for C(, w) just found and (5) we have G(, w) = 1 + G() + wg() wc(g()) 1 y C (G())w = 1 + w + ( + w w)g(), 1 y G ()w where y G () = (G() 1)/. In all three cases, the condition from Lemma 1.3 on the numerator of B(, w), C(, w), and G(, w) being analytic (as a function of 1 / 0 ) in the region D follows from the fact that C() and G() have singularities of the square-root type. Next we consider the generating functions B(, w, t), C(, w, t), and G(, w, t). Recall that, in addition to the degree of the root p 1, the degree of a secondary vertex p j (with j 1) is marked by the variable t. For -connected graphs we consider two different cases. The generating function B 1 (, w, t) deals with the degree of p and the generating function B (, w, t) with the general case p j, j 3. These functions were already computed in [10], as follows: B 1 (, w, t) = wt + B (, w, t) = w t (1 + (A + 1)) (1 w(a + 1))(1 t(a + 1)) w t (1 + (A + 1))(P 1 + (wt w t)p ) (1 (4A + 3))(1 w(a + 1)) (1 t(a + 1)), P 1 = 1 (4A + 1), P = 1 A + (A + 1), A = By definition, B(, w, t) = B 1 (, w, t)+b (, w, t). Notice that (A+1) = ( )/ is precisely y B (). The factor 1 (4A + 3) in the denominator contributes precisely to the factor Z = 1 / 0 (with 0 = 3 ) in the denominator of (1). Hence the conditions of Lemma 1.3 are satisfied. For connected graphs we also consider two different situations. We write C(, w, t) = C 1 (, w, t)+ C (, w, t), where the generating function C 1 (, w, t) deals with the case when the secondary root is in one of the blocks that contain the root vertex. We claim C 1 (, w, t) = C(, (1 B(C /, w)) B(C t) /, w, t) C. Indeed, the first factor corresponds to the blocks that do not contain the secondary root, the middle factor to the block that contains it, and the last factor corresponds to the two connected graphs attached to the secondary root. For C (, w, t) the secondary root lies in one of the connected graphs that are attached to the blocks that contain the root. We have C (, w, t) = (1 B(C /, w)) B (C C(, 1, t)c /, w), where the middle factor identifies a block and a vertex v in that block, and the last factor picks a vertex in one of the connected graphs attached to v to become the secondary root. Observe that the term /(1 B(C /)) equals C(, w) /, hence ( + w wc) (1 B(C = /, w)) (1 y C ()w). (1) 10

11 For the term C(, 1, t), notice that if C(, t) = c n,k n t k, then C(, 1, t) = (n 1)c n,k n t k, hence C(, 1, t) = C (, t) C(, t). From (4) we obtain C () = ( 4 + 3C)(3C + C 3) 1, which we can substitute in the derivative of C(, t) to find the following expression for C(, 1, t): C(, 1, t) = tc ( (1 t)c + 4tC t ) (3C + C 3)(1 ty C ()). () The term B (C /, w) can be expressed in terms of C in a similar way, first using (1) to find B (, w) in terms of, w and B(), and then using (3) to obtain B (C /, w) = w ( C + w(c C + )) ( C )( w(c )). Finally, for the term B(C /, w, t) we need A(C /) = ( 3C 6C + C 4 )/(4C ). Eliminating from (4), one can check that the term inside the square root is a perfect square, namely, 6C + C 4 = (C + C 3). It turns out that the correct square root is (C + C 3), therefore A(C /) = C C C. Substituting into B 1 (, w, t) and B (, w, t), we obtain the following expression, which is enough for our purposes. B(C /, w, t) = P (C,, w, t) (C (C + C ))(C w(c )) (C t(c )), for an explicit polynomial P (c,, w, t). It remains to check that C(, w, t) has the form given in (1). The terms (1 wy C ()) and (1 ty C ()) in the denominator appear clearly from (1) and (). Among all other terms, one checks that the only one that contributes to a factor Z = 1 / 0 (now 0 = 3/18) is the term 3C + C 3 from (). This follows from the expansion of C() in powers of Z, which is ( ) C() = 6 9 Z + O(Z ). 6 Finally, for arbitrary graphs we have G(, w, t) = 1 + C(G(), w, t) + C (G(), w)g (, t). The second summand corresponds to the case where the secondary vertex is in the same component as the root. The third summand corresponds to the opposite case, and the derivative C marks the component containing the secondary vertex; the term G (, t) marks the secondary vertex. To find G (, t) we proceed as we did for C (, t) above and find G (, t) = Q(G(),, t) (G() + 3 )(1 ty G ()) for a polynomial Q(g,, t). The substitution of = G() in the already known terms C(, w, t) and C (, w) is straightforward since C(G()) = G() 1. The terms (1 wy C (G()) and (1 ty C (G()) give rise to the required terms (1 wy G ()) and (1 ty G ()) ; it is routine to check that among all other terms, the only one that contributes a factor 1 / 0 in the denominator (with 0 = 3/ ) is the term G() + 3 from G (, t). 11

12 4 Sie of the largest component The sie M n of the largest component can be handled with the same tools as the maximum degree, but we need to refine the analysis. Let X n,k denote the number of components of sie k in a random graph of sie n and set Y n,k = l>k X n,l the number of components with more than k vertices. Then we have Y n,k > 0 M n > k. Hence, by applying the first and second moment method we can estimate the probabilities P{M n > k} with the help of the first two moments E Y n,k and E Yn,k. By definition we have X n,k = 1 n 1 [ {component of pi} =k] k and consequently i=1 E X n,k = n k q n,k, where q n,k denotes the probability that the root component has sie k. By definition q n,k g n is the number of graphs where the root component has sie k; recall that g n denotes the number of graphs of sie n. Hence the the corresponding generating function is given by G(, u) = n,k q n,k g n n u k = 1 + C(uG()). (3) From this we obtain the following property. Lemma 4.1. There exist constants c > 0 and ρ < 1 such that uniformly for k C log n (where C > 0 is an arbitrary constant) E Y n,k c nρ k k 3/. (4) Furthermore we have for every ε > 0 uniformly for all n, k 0. E Y n,k = O ( n(ρ + ε) k) (5) Proof. Set r() = G()/ρ C, where ρ C = 3/18 denotes the singularity of C(). Then r() has a square-root singularity at = ρ G = 3/. Setting Z = 1 /ρ G we can, thus, represent G(, u) locally as G(, u) = g 1 (, u, Z) h 1 (, u, Z) 1 ur(), where g 1 and h 1 are analytic functions at (, u, Z) = (ρ G, u 0, 0), u 0 = ρ C /(ρ G G(ρ G )) > 1, and h 1 is non-ero at this point. Now we use the methods of [10, Appendix B] to obtain an asymptotic expansion of the form [ n u k ] G(, u) c 1 ρ n G n 3/ u k 0 k 1/ for a certain constant c 1 > 0 (that is uniform for k C log n, where C > 0 is an arbitrary constant). Of course this implies E X n,k c nρ k k 3/. for some constant c > 0 and with ρ = 1/u 0. Furthermore by using the inequality we obtain for every ε > 0 [ n u k ] G(, u) (ρ + ε) k [ n ]G(, (ρ + ε) 1 ), E X n,k = O ( n(ρ + ε) k). Clearly these two properties of E X n,k imply (4) and (5). 1

13 The estimates (4) and (5) imply an upper bound for E M n of the form log n/ log(1/ρ). A corresponding lower bound follows by considering the second moment. Since we obtain X n,k = 1 k n 1 [ {component of pi} =k]1 [ {component of pj} =k] i,j=1 E X n,k = n k n q n,k,k;j where q n,k,k;j denotes the probability that the component of the root p 1 as well as the component of p j have sie k. Similarly we have for k l j=1 E X n,k X n,l = n kl n j=1 q n,k,l;j where q n,k,l;j denotes the probability that the component of p 1 has sie k and the component of p j has sie l. Note that q n,k,l;1 = 0 if k l but for the sake of consistency we include this term in the formula. If we know the behaviour of E X n,k X n,l, we also obtain that of E Yn,k = E X n,l1 X n,l. l 1,l >k In order to deal with these second moments we introduce another variable v that takes care of the sie of a component of a vertex p j, 1 j n. The corresponding generating function is given by G(, u, v) = n q n,k,l;j g n x n u k v l n,k,l j=1 and satisfies the relation Consequently it is also given by G(, u, v) = 1 + C (uvg())uvg() + C (ug())ug(, 1, v). G(, u, v) = 1 + C (uvg())uvg() + C (ug())u 1 + C (vg())vg() 1 C. (G()) This representation leads to the following property for the second moment. Lemma 4.. We have uniformly for k C log n and l C log n (where C > 0 is an arbitrary constant) E Y n,k c n ρ k k 3 (1 + o(1)), (6) where the constant c is the same as that of Lemma 4.1. Proof. It is now an easy exercise to show that the asymptotic leading term of G(, u, v) can be represented as C (ug())uc (vg())vg() 1 C (G()) = H(, Z, u, v) Z 1 ur() 1 vr(), where H(, x, u, v) is non-ero and regular at (, x, u, v) = ( 0, 0, g( 0 ), g( 0 )). Again we use the methods of [10] to obtain asymptotic expansions of the form c n 1/ ρ n G ρk+l (kl) 1/ (1 + o(1)) 13

14 for the coefficients. This leads to an asymptotic expansion for E X n,k X n,l of the form E X n,k X n,l = c n ρ k+l (kl) 3/ (1 + o(1)). Similarly to the calculations of Lemma 4.1 we also get a uniform upper bound of the form E X n,k X n,l = O ( n (ρ + ε) k+l) (for every ε > 0). Consequently the asymptotic expansion (6) for E Yn,k constant. follows, and it is easy to check the With the help of Lemmas 4.1 and 4. we obtain corresponding lower bounds for E M n. To state our main result we use the exact value of ρ. Theorem 4.3. The sie M n of the largest component in non-crossing graphs satisfies M n log n c in probability, where c = 1/ log(3 3(49 69)/). We also have E M n c log n as n. A similar result holds for the sie of the largest -connected component in connected graphs. We omit the details for the sake of conciseness. 5 Diameter The diameter of a connected graph is the maximum distance between any pair of vertices. In this section we show that the diameter D n of connected and -connected non-crossing graphs is of order n. To get precise estimates on the diameter in this context is usually hard (to our knowledge, only in the case of trees exact results are known [17, 3]). Instead, we prove results on the parameter d n, which is the maximum distance from a vertex to the root p 1. Since clearly we have d n D n d n, it is enough to obtain bounds of the right order of magnitude for d n. In what follows, all the results are stated in terms of d n, and the reader should keep in mind that they provide upper and lower bounds for the diameter D n of the same order of magnitude. Before we discuss -connected graphs we consider the special case of triangulations of a convex polygon, which is interesting in itself. As it is well-known, the number( of triangulations of a convex n-gon (where the root p 1 is not counted) is the Catalan number 1 n ) n n 1. The associated generating function T () satisfies the equation and has the explicit solution T () = + T (), T () = Lemma 5.1. Let T k () be the generating function of triangulations of a convex n-gon with d n k. Then T 0 () = 0 and T k () = 1 T k 1 () (k 1). (7) 14

15 Proof. Let T k,l () be the generating function of triangulations of a convex n-gon, where for each vertex the distance to the root vertex p 1 is at most k or the distance to the vertex p n is at most l, where we require moreover that the edge (p 1, p n ) is not used in computing these distances. It is clear that T k () = T k,k 1 (). By considering the two triangulations adjacent to the root face we have the recurrence relations which lead directly to the recurrence T k,k () = + T k,k 1 (), T k,k 1 () = + T k,k 1 ()T k 1,k 1 (), T k () = 1 T k 1 (). Theorem 5.. The expected value of d n in triangulations is asymptotically given by E d n 3 πn. Proof. We apply Lemma 1.4 with F (, t) = /(1 t ) and the parameters 0 = 1 4, t 0 = 1, F ( 0, t 0 ) = 3, F tt ( 0, t 0 ) = 6. Clearly 1/4 is the singularity of T (), and t 0 = T (1/4) = 1/. The remaining relations are immediate Connected graphs For the analysis of -connected graphs we need a combinatorial decomposition different from the one we have used so far. Consider the root region of a -connected graph and let p j be the vertex that follows p 1 when traversing the root region counterclockwise (see Figure 3). The vertices p 1,..., p j induce a -connected graph, whereas the vertices p j,..., p n induced either a -connected graph or a graph that is -connected after the addition of the edge p j p n. This leads to the equation B() = + B()(B() ). p 1 p n (1) () p 3 p 1 () p 1 (4) () (3) p j (3) p 5 () p 7 (1) p 8 (1) () Figure 3: Decomposition of a -connected graph (left) and of a connected graph (right), where the numbers in parentheses indicate the value of d for each non-root vertex. Lemma 5.3. Let B k () be the generating function of -connected graphs with d n k Then B 0 () = 0 and B k () = 1 4B k 1 () + B k 1 () (k 1). (8) 15

16 Proof. Let B k,l () be defined in the same way as in the proof of Lemma 5.1. recurrence relations We claim the B k,k 1 () = + B k,k 1 ()(B k 1,k 1 () ) B k 1,k 1 () = + B k 1,k ()(B k 1,k () ) from which the statement of the lemma follows directly. Both relations follow by looking at distances in the two -connected graphs that arise in the previous decomposition. Theorem 5.4. The expected value of d n in -connected graphs is given asymptotically by with c = (3 + ) 1/4 / E d n c πn. Proof. We apply Lemma 1.4 with F (, t) = (1 4t + t) 1 and the parameters 0 = 3, t 0 = 1, F ( 0, t 0 ) = 1 +, F tt ( 0, t 0 ) = 8 +. This is correct since 0 is the singularity of B() and B(3 ) = 1 /. 5. Connected graphs The diameter in connected graphs is also of order n. As above we only discuss the largest distance d n to the root vertex p 1, but instead of studying d n directly, we look at two new parameters d n and d n that satisfy d n d n d n. For the lower bound, we consider the tree structure of the block decomposition of a connected graph. Let d n be the maximum number of cut-points on a path to the root vertex (where the root vertex is never counted as a cut-point). Lemma 5.5. Let C k () be the generating function corresponding to those connected graphs with d n k. Then we have C 0 () = /(1 B()) and C k () = 1 B(C k 1 (), (k 1). /) Proof. This is immediate from the block decomposition of connected graphs. Lemma 5.6. The expected value of d n is asymptotically given by with c = (1 3/3) E d n c πn. Proof. By combining the recurrence in Lemma 5.5 and Equation (), we find the following explicit expression for C k () in terms of C k 1 (): C k () = 3 1 C k 1 1 6Ck 1 + C4 k 1. The result follows by applying Lemma 1.4 with F (, t) = 3/ t / 6t + t 4 / and parameters 3 0 = 18, t 0 = 1 6 ( 3 1), F ( 0, t 0 ) = 3 3, F tt ( 0, t 0 ) = 9( ). 16

17 For the upper bound, we use an alternative decomposition of connected graphs [11]. Take the root of a connected graph and let p i1,..., p id be its neighbours, with i 1 < i < < i d. The subgraph induced by {p ij, p ij+1,..., p ij+1 } is either connected or it has exactly two connected components. Also, the subgraphs induced by {p, p 3,..., p i1 } and {p id, p id +1,..., p n } are connected. (See Figure 3.) This decomposition produces the equation ( C() ) C() = 1 + C() C(). (9) Now define an application d G from the set of vertices of a connected graph G to N recursively as follows. If x is the root-vertex, then d G (x) = 0. Otherwise, the vertex x belongs, according to the decomposition scheme, to at least one connected subgraph C which has either one or two vertices that are neighbours of the root. If x is adjacent to the root of G, then x belongs to two such subgraphs (which could be reduced to a single vertex); in this case, let C be the one that contains the vertex with smallest label. Define d G (x) = d C (x) + 1, where the root of C is taken to be the vertex with smallest label among those adjacent to the root of G. For instance, in the graph in Figure 3, the values of d are indicated for each vertex. Let d n be the maximum of d G in a connected graph G with n vertices. Clearly d n is an upper bound for d n. The following lemma is immediate from the alternative decomposition of connected graphs. Lemma 5.7. Let C k () be the generating function corresponding to those connected non-crossing graphs with d n k. Then we have C 0 () = and ( C k () = 1 + C k 1 () C k 1 () C k 1 () Lemma 5.8. The expected value of d n is asymptotically given by with c = (1 + 3/3)/ E d n c πn, ), (k 1). Proof. We apply Lemma 1.4 for F (, t) = (1 + t /( t t )) and parameters 3 0 = 18, t 0 = 1 6 ( 3 1), F ( 0, t 0 ) = 9 5 3, F tt ( 0, t 0 ) = 18(1 + 3). Corollary 5.9. The expected value of d n in connected graphs satisfies for suitable constants c 1 and c. 6 Concluding remarks c 1 n E dn c n One of the motivations for this work was to solve an open problem in [15], namely to count bipartite non-crossing graphs. This can be solved by observing that a dissection is a bipartite graph if and only if all the regions have even sie, and an arbitrary graph is bipartite if and only if all its components and blocks are bipartite. Let B b (), C b () and G b () be the corresponding generating functions for bipartite configurations. The relations between G b () and C b () and between C b () and B b (), are given by the 17

18 analogous of Equations (5) and (3). To find the equation for B b (), we modify Equation (1) by allowing only cycles of even length in the decomposition of dissections. This gives rise to This determines everything and we obtain B b () = + B b() 3 1 B b (). B b () = C b () = G b () = Asymptotics can be obtained too and we have [ n ]B b () c 1 n 3/ γ n 1, γ , [ n ]C b () c n 3/ γ n, γ 7.589, [ n ]G b () c 3 n 3/ γ n 3, γ As a final remark, it is shown in [8] that for every k 1 the random variable that counts the number of vertices of degree k in dissections is asymptotically normal with linear expectation and variance (corresponding to -connected outerplanar graphs). With the same techniques, based on multivariate functional equations, it is possible to prove the same result for connected and arbitrary non-crossing graphs. References [1] N. Alon, J. H. Spencer, The probabilistic method, third edition, John Wiley & Sons, Inc., Hoboken, NJ, 008. [] N. Bernasconi, K. Panagiotou, A. Steger, On properties of random dissections and triangulations, Combinatorica 30 (0), [3] N. Broutin, P. Flajolet, The distribution of height and diameter in random non-plane binary trees, Random Structures Algorithms 41 (), [4] L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, [5] E. Deutsch, M. Noy, Statistics on non-crossing trees, Discrete Math. 54 (00), [6] L. Devroye, P. Flajolet, F. Hurtado, M. Noy, W. Steiger, Properties of random triangulations and trees, Discrete Comput. Geom. (1999), [7] M. Drmota, Random Trees, Springer, Wien-New York, 009. [8] M. Drmota, O. Giméne, M. Noy, Vertices of given degree in series-parallel graphs, Random Structures Algorithms 36 (0), [9] M. Drmota, O. Giméne, M. Noy, Degree distribution in random planar graphs, J. Combin. Theory Ser. A 118 (1), [10] M. Drmota, O. Giméne, M. Noy, The maximum degree of series-parallel graphs, Combin. Probab. Comput. 0 (1), [11] P. Flajolet, M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math. 04 (1999),

19 [1] P. Flajolet, A. Odlyko, The average height of binary trees and other simple trees. J. Comput. System Sci. 5 (198), [13] P. Flajolet, R. Sedgewick, Analytic combinatorics, Cambridge U. Press, Cambridge, 009. [14] Z. Gao, N. C. Wormald, The distribution of the maximum vertex degree in random planar maps, J. Combin. Theory Ser. A 89 (000), 30. [15] C. Huemer, Structural and Enumerative Problems for Plane Geometric Graphs, Ph.D. Thesis, Universitat Politècnica de Catalunya, 007. [16] A. Panholer, The height distribution of nodes in non-crossing trees, Ars Combin. 69 (003), [17] G. Sekeres, Distribution of labelled trees by diameter, Lecture Notes in Math (1983),