Asymptotic properties of some minor-closed classes of graphs
AASYMPTOTIC PROPERTIES OF SOMEMINOR-CLOSED CLASSES OF GRAPHS
MIREILLE BOUSQUET-MÉLOU AND KERSTIN WELLER
Abstract.
Let A be a minor-closed class of labelled graphs, and let G n be a random graphsampled uniformly from the set of n -vertex graphs of A . When n is large, what is theprobability that G n is connected? How many components does it have? How large is itsbiggest component? Thanks to the work of McDiarmid and his collaborators, these questionsare now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes A excluding non-2-connected mi-nors, and show that their asymptotic behaviour may be rather different from the 2-connectedcase. This behaviour largely depends on the nature of dominant singularity of the generatingfunction C ( z ) that counts connected graphs of A . We classify our examples accordingly, thustaking a first step towards a classification of minor-closed classes of graphs. Furthermore, weinvestigate a parameter that has not received any attention in this context yet: the size of theroot component. It follows non-gaussian limit laws (beta and gamma), and clearly deserves asystematic investigation. Introduction
We consider simple graphs on the vertex set { , . . . , n } . A set of graphs is a class if it isclosed under isomorphisms. A class of graphs A is minor-closed if any minor of a graph of A isin A . To each such class one can associate its set E of excluded minors : an (unlabelled) graphis excluded if its labelled versions do not belong to A , but the labelled versions of each of itsproper minors belong to A . A remarkable result of Robertson and Seymour states that E isalways finite [32]. We say that the graphs of A avoid the graphs of E . We refer to [6] for a studyof the possible growth rates of minor-closed classes.For a minor-closed class A , we study the asymptotic properties of a random graph G n takenuniformly in A n , the set of graphs of A having n vertices: what is the probability p n that G n isconnected? More generally, what is the number N n of connected components? What is the size S n of the root component , that is, the component containing 1? Or the size L n of the largestcomponent?Thanks to the work of McDiarmid and his collaborators, a lot is known if all excluded graphsare 2-connected: then p n converges to a positive constant (at least / √ e ), N n converges in lawto a Poisson distribution, n − S n and n − L n converge in law to the same discrete distribution.Details are given in Section 3.If some excluded minors are not G n may be rather different(imagine we exclude the one edge graph...). This paper takes a preliminary step towards aclassification of the possible behaviours by presenting an organized catalogue of examples.For each class A that we study, we first determine the generating functions C ( z ) and A ( z ) thatcount connected and general graphs of A , respectively. The minors that we exclude are alwaysconnected , which implies that A is decomposable in the sense of Kolchin [24]: a graph belongs to A if and only if all its connected components belong to A . This implies that A ( z ) = exp( C ( z )) . Date : November 1, 2018.Both authors were partially supported by a CNRS-Oxford collaboration scheme (CNRS/John Fell OxfordUniversity Press Research fund, 2012). obtained by contracting or deleting some edges, removing some isolated vertices and discarding loops andmultiple edges We refer to [27] for an example where this is not the case. a r X i v : . [ m a t h . C O ] F e b M. BOUSQUET-MÉLOU AND K. WELLER
Figure 1.
A zoo of graphs. Top: the 3-star, the triangle K , the bowtie andthe diamond. Bottom: A caterpillar and the 4-spoon (a k -spoon consists of a“handle” formed of k edges, to which a triangle is attached).We then derive asymptotic results from the values of these series. They are illustrated throughoutthe paper by pictures of large random graphs, generated using Boltzmann samplers [18]. Under aBoltzmann distribution, two graphs of A having the same size always have the same probability.The most difficult class we study is that of graphs avoiding the bowtie (shown in Figure 1).Our results make extensive use of the techniques of Flajolet and Sedgewick’s book [19]: sym-bolic combinatorics, singularity analysis, saddle point method, and their application to thederivation of limit laws. We recall a few basic principles in Section 2. We also need and provetwo general results of independent interest related to the saddle point method or, more precisely,to Hayman admissibility (Theorems 17 and 18).Our results are summarized in Table 1. A first principle seems to emerge: the more rapidly C ( z ) diverges at its radius of convergence ρ , the more compo-nents G n has, and the smaller they are. In particular, when C ( ρ ) converges, then the properties of G n are qualitatively the same as inthe 2-connected case (for which C ( ρ ) always converges [26]), except that the limit of p n can bearbitrarily small. When C ( ρ ) diverges, a whole variety of behaviours can be observed, dependingon the nature of the singularity of C ( z ) at ρ : the probability p n tends always to , but at variousspeeds; the number N n of components goes to infinity at various speeds (but is invariablygaussian after normalization); the size S n of the root component and the size L n of the largestcomponent follow, after normalization, non-gaussian limit laws: for instance, a Gamma or Betalaw for S n , and for L n a Gumbel law or the first component of a Poisson-Dirichlet distribution.Cases where C ( z ) converges, or diverges at most logarithmically, are addressed using singularityanalysis (Sections 4 and 5), while those in which C ( z ) diverges faster (in practise, with analgebraic singularity) are addressed with the saddle point method (Sections 7 to 10). Section 6gathers general results on the saddle point method and Hayman admissibility.Let us conclude with a few words on the size of the root component. It appears that thisparameter, which can be defined for any exponential family of objects, has not been studiedsystematically yet, and follows interesting ( i.e. , non-gaussian!) continuous limit laws, after nor-malization. In an independent paper [10], we perform such a systematic study, in the spirit ofwhat Bell et al . [4] or Gourdon [21] did for the number of components or the largest compo-nent, respectively. This project is also reminiscent of the study of the 2-connected componentcontaining the root vertex in a planar map, which also leads to a non-gaussian continuous limitlaw, namely an Airy distribution [3]. This distribution is also related to the size of the largest2- and 3-connected components in various classes of graphs [20].2. “Generatingfunctionology” for graphs Let E be a finite set of (unlabelled) connected graphs that forms an antichain for the minororder (this means that no graph of E is a minor of another one). Let A be the set of labelledgraphs that do not contain any element of E as a minor. We denote by A n the subset of A formed of graphs having n vertices (or size n ) and by a n the cardinality of A n . The associatedexponential generating function is A ( z ) = (cid:80) n ≥ a n z n /n ! . We use similar notation ( c n and C ( z ) ) SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 3 for the subset C of A consisting of (non-empty) connected graphs. Since the excluded minorsare connected, A is decomposable, and A ( z ) = exp( C ( z )) . Several refinements of this series are of interest, for instance the generating function that keepstrack of the number of (connected) components as well: A ( z, u ) = (cid:88) G ∈A u c ( G ) z | G | | G | ! , where | G | is the size of G and c ( G ) the number of its components. Of course, A ( z, u ) = exp( uC ( z )) . Excluded C ( ρ ) Sing. lim p n number N n root largest Refs. andminors of C ( z ) of comp. comp. S n comp. L n methods2-connected < ∞ ? ≥ / √ e O (1) n − S n n − L n [1, 26, 28, 29] < Poisson → disc. → disc. Sec. 3at least < ∞ (1 − ze ) / > id. id. id. Sec. 4a spoon, ≤ / √ e sing. an.but no tree ∞ log log n n PD (1) (1 / Sec. 5( + √ ) gaussian (1 − x ) − / sing. an. ∞ / √ n / n / ? Sec. 10gaussian (cid:112) x/πe − x saddle ∞ simple 0 √ n √ n √ n log n Sec. 8(path forests) pole gaussian xe − x Gumbel saddle ∞ id. 0 id. id. ? Sec. 8(forests of saddlecaterpillars) ∞ id. 0 id. id. ? Sec. 9(max. deg. 2) ( + log ) saddleall conn. graphs ∞ entire n/k k k Sec. 7of size k + 1 (polynomial) gaussian Dirac Dirac saddle Table 1.
Summary of the results: for each quantity N n , S n and L n , we give theorder of the expected value (up to a multiplicative constant, except in the lastline where constants are exact) and a description (name or density) of the limitlaw. The examples are ordered according to the speed of divergence of C ( z ) near its radius ρ . Spoons are defined in Figure 1. As we get lower in the table,the graphs have more components, of a smaller size. The symbol PD (1) (1 / stands for the first component of a Poisson-Dirichlet distribution of parameter / . M. BOUSQUET-MÉLOU AND K. WELLER
We denote by G n a uniform random graph of A n , and by N n the number of its components.Clearly, P ( N n = i ) = [ z n ] C ( z ) i i ![ z n ] A ( z ) , (1)where [ z n ] F ( z ) denotes the coefficient of z n in the series F ( z ) . The i th factorial moment of N n is E ( N n ( N n − · · · ( N n − i + 1)) = [ z n ] ∂ i A∂u i ( z, z n ] A ( z ) = [ z n ] C ( z ) i A ( z )[ z n ] A ( z ) . Several general results provide a limit law for N n if C ( z ) satisfies certain conditions: for instancethe results of Bell et al . [4] that require C ( z ) to converge at its radius of convergence; or the exp-log schema of [19, Prop. IX.14, p. 670], which requires C ( z ) to diverge with a logarithmicsingularity (see also the closely related results of [2] on logarithmic structures). We use theseresults when applicable, and prove a new result of this type, based on Drmota et al .’s notion of extended Hayman admissibility , which applies when C ( z ) diverges with an algebraic singularity.We believe it to be of independent interest (Theorem 18).We also study the size c of the root component , which is the component containing thevertex 1. We define accordingly ¯ A ( z, v ) = (cid:88) G ∈A ,G (cid:54) = ∅ v c ( G ) − z | G |− ( | G | − . The choice of | G | − instead of | G | simplifies slightly some calculations. Note that ¯ A ( z,
1) = A (cid:48) ( z ) = C (cid:48) ( z ) A ( z ) . Denoting by S n the size of the root component in G n , we have P ( S n = k ) = c k a n − k (cid:0) n − k − (cid:1) a n = kn c k k ! a n − k ( n − k )! n ! a n . (2)Equivalently, the series ¯ A ( z, v ) is given by ¯ A ( z, v ) = C (cid:48) ( zv ) A ( z ) . (3)The i th factorial moment of S n − is E (( S n − · · · ( S n − i )) = [ z n − ] ∂ i ¯ A∂v i ( z, z n − ] ¯ A ( z,
1) = [ z n − i − ] C ( i +1) ( z ) A ( z ) n [ z n ] A ( z ) . (4)Surprisingly, this parameter has not been studied before. Our examples give rise to non-gaussianlimit laws (Beta or Gamma, cf. Propositions 14 or 25). In fact, the form (3) of the generatingfunction shows that this parameter is bound to give rise to interesting limit laws, as both thelocation and nature of the singularity change as v moves from − ε to ε . Using the terminologyof Flajolet and Sedgewick [19, Sec. IX.11], a phase transition occurs. We are currently workingon a systematic study of this parameter in exponential structures [10].Finally, we denote by C [ k ] ( z ) the generating function of connected graphs of A of size lessthan k : C [ k ] ( z ) = k − (cid:88) n =1 c n z n n ! , and study, for some classes of graphs, the size L n of the largest component. We have P ( L n < k ) = [ z n ] exp( C [ k ] ( z ))[ z n ] A ( z ) . (5)We use in this paper two main methods for studying the asymptotic behaviour of a sequence ( a n ) n given by its generating function A ( z ) . The first one is the singularity analysis of [19,Chap. VI]. Let us describe briefly how it applies, for the readers who would not be familiar with SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 5 it. Assume that A ( z ) has a unique singularity of minimal modulus (also called dominant ) at itsradius of convergence ρ , and is analytic in a ∆ -domain , that is, a domain of the form { z : | z | < r, z (cid:54) = ρ and | Arg( z − ρ ) | > φ } for some r > ρ and φ ∈ (0 , π/ . Assume finally that, as z approaches ρ in this domain, A ( z ) = S ( z ) + O ( R ( z )) , where S ( z ) and R ( z ) are functions belonging to the simple algebraic-logarithmic scale of [19,Sec. VI.2]. Then one can transfer the above singular estimates for the series into asymptoticestimates for the coefficients: [ z n ] A ( z ) = [ z n ] S ( z ) + O ([ z n ] R ( z )) . Since S and R are simple functions, the asymptotic behaviour of their coefficients is well known,and the estimate of [ z n ] A ( z ) is thus explicit. We use singularity analysis in Sections 3 to 5. Thesecond method we use is the saddle point method . In Section 6 we recall how to apply it, andthen use it in Sections 7 to 10.When dealing directly with sequences rather than generating functions, a useful notion willbe that of smoothness : the sequence ( f n ) n ≥ is smooth if f n − /f n converges as n grows. Thelimit is then the radius of convergence of the series (cid:80) n f n z n .3. Classes defined by 2-connected excluded minors
We assume in this section that at least one minor is excluded, and that all excluded minorsare 2-connected. This includes the classes of forests, series-parallel graphs, outer-planar graphs,planar graphs... Many results are known in this case. We recall briefly some of them, and statea new (but easy) result dealing with the size of the root component. The general picture is thatthe class A shares many properties with the class of forests. Proposition 1 ( The number of graphs — when excluded minors are 2-connected).
Thegenerating functions C ( z ) and A ( z ) = e C ( z ) are finite at their (positive) radius of convergence ρ .Moreover, the sequence ( a n /n !) n is smooth.The probability that G n is connected tends to /A ( ρ ) , which is clearly in (0 , . In fact, thislimit is also larger than or equal to / √ e . The latter value is reached when A is the class offorests. The fact that ρ is positive is due to Norine et al . [30], and holds for any proper minor-closedclass. The next results are due to McDiarmid [26] (see also the earlier papers [28, 29]). The factthat /A ( ρ ) ≥ / √ e , or equivalently, that C ( ρ ) ≤ / , was conjectured in [29], and then provedindependently in [1] and [23]. Example 2.
A basic, but important example is that of forests, illustrated in Figure 2. We havein this case C ( z ) = T ( z ) − T ( z ) , where T ( z ) = ze T ( z ) counts rooted trees (see for instance [19, p. 132]). The series T , C and A = e C have radius of convergence ρ = 1 /e , with the following singular expansions at this point: T ( z ) = 1 −√ − ze ) / + (1 − ze ) − √ (1 − ze ) / + O ((1 − ze ) ) ,C ( z ) = − (1 − ze ) + √ (1 − ze ) / + O ((1 − ze ) ) ,A ( z ) = √ e −√ e (1 − ze ) + √ e √ (1 − ze ) / + O ((1 − ze ) ) . (6)The singularity analysis of [19, Chap. VI] applies: the three series are analytic in a ∆ -domain,and their coefficients satisfy t n ∼ n ! e n √ πn / , c n ∼ n ! e n √ πn / , and a n ∼ √ e c n . M. BOUSQUET-MÉLOU AND K. WELLER
Figure 2.
A random forest of size n = 1165 . It has two connected components.We will also consider rooted trees of height less than k (where by convention the tree consistingof a single vertex has height ). Let T k ( z ) denote their generating function. Then T ( z ) = z andfor k ≥ , T k +1 ( z ) = ze T k ( z ) . Note that T k ( z ) is entire. Note.
When all excluded minors are 2-connected, C ( ρ ) always converges, but the nature of thesingularity of C ( z ) at ρ depends on the class: it is for instance (1 − z/ρ ) / for forests (and moregenerally, for subcritical classes [14]), but (1 − z/ρ ) / for planar graphs. We refer to [20] for amore detailed discussion that applies to classes that exclude 3-connected minors. Proposition 3 ( Number of components — when excluded minors are 2-connected).
The mean of N n satisfies: E ( N n ) ∼ C ( ρ ) and the random variable N n − converges in law to a Poisson distribution of parameter C ( ρ ) .That is, as n → ∞ , P ( N n = i + 1) → C ( ρ ) i i ! e C ( ρ ) . (7)We refer to [26, Cor. 1.6] for a proof. The largest component is known to contain almost allvertices, and it is not hard to prove that the same holds for the root component. In fact, thetails of the random variables S n and L n are related by the following simple result. Lemma 4.
For any class of graphs A , and k < n/ , P ( S n = n − k ) = n − kn P ( L n = n − k ) . SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 7
Proof.
Let us denote by B n the (lexicographically first) biggest component of G n . Its size is thus L n . We have, for n > k , P ( S n = n − k ) = P ( S n = n − k and ∈ B n ) + P ( S n = n − k and (cid:54)∈ B n )= P ( L n = n − k and ∈ B n ) + P ( S n = n − k and (cid:54)∈ B n )= P (1 ∈ B n | L n = n − k ) P ( L n = n − k ) + P ( S n = n − k and (cid:54)∈ B n )= n − kn P ( L n = n − k ) . Indeed, there cannot be two components of size n − k or more. This implies that P ( S n = n − k and (cid:54)∈ B n ) = 0 . Proposition 5 ( The root component and the largest component — when excludedminors are 2-connected).
The random variables n − S n and n − L n both converge to a discretelimit distribution X given by P ( X = k ) = 1 A ( ρ ) a k ρ k k ! . Proof.
By Lemma 4, the two statements are equivalent. The L n result has been proved byMcDiarmid [26, Cor. 1.6].We give an independent proof (of the S n result), as we will recycle its ingredients later forcertain classes of graphs that avoid non-2-connected minors. Let k ≥ be fixed. By (2), P ( S n = n − k ) = c n − k a k (cid:0) n − k (cid:1) a n = a k k ! c n − k a n − k ( n − a n − k ( n − k − a n . By Proposition 1, the term c n − k /a n − k , which is the probability that a graph of size n − k isconnected, converges to /A ( ρ ) . Moreover, the sequence a n /n ! is smooth, so that ( n − a n − k ( n − k − a n converges to ρ k . The result follows.A more precise result is actually available. Let us call fragment the union of the componentsthat differ from the biggest component B n . Then McDiarmid describes the limit law of thefragment, not only of his size [26, Thm. 1.5]: the probability that the fragment is isomorphic toa given unlabelled graph H of size k is A ( ρ ) ρ k aut( H ) , where aut( H ) is the number of automorphisms of H .4. When trees dominate: C ( z ) converges at ρ Let A be a decomposable class of graphs (for instance, a class defined by excluding connectedminors), satisfying the following conditions:(1) A includes all trees,(2) the generating function D ( z ) that counts the connected graphs of A that are not treeshas radius of convergence (strictly) larger than /e (which is the radius of trees).We then say that A is dominated by trees . Some examples are presented below. In this case, theproperties that hold for forests (Section 3) still hold, except that the probability c n /a n that G n is connected tends to a limit that is now at most / √ e . We will see that this limit can becomearbitrarily small. Proposition 6 ( The number of graphs — when trees dominate).
Let T ( z ) be the gener-ating function of rooted trees, given by T ( z ) = ze T ( z ) . Write the generating function of connectedgraphs in the class A as C ( z ) = T ( z ) − T ( z ) D ( z ) . M. BOUSQUET-MÉLOU AND K. WELLER
The generating function of graphs of A is A ( z ) = e C ( z ) . As n → ∞ , c n ∼ n ! e n √ πn / and a n ∼ A (1 /e ) c n . In particular, the probability that G n is connected tends to /A (1 /e ) = e − / − D (1 /e ) as n → ∞ .Proof. As in Example 2, we use singularity analysis [19, Chap. VI]. By assumption, D ( z ) hasradius of convergence larger than /e , and the singular behaviour of C ( z ) is that of unrootedtrees. More precisely, it follows from (6) that, as z approaches /e , C ( z ) = 1 / D (1 /e ) − (1 − ze )(1 + D (cid:48) (1 /e ) /e ) + 2 √
23 (1 − ze ) / + O ((1 − ze ) ) , this expansion being valid in a ∆ -domain. This gives the estimate of c n via singularity analysis.For the series A , we find A ( z ) = e / D (1 /e ) (cid:32) − (1 − ze )(1 + D (cid:48) (1 /e ) /e ) + 2 √
23 (1 − ze ) / + O ((1 − ze ) ) (cid:33) , and the estimate of a n follows. Proposition 7 ( Number of components — when trees dominate).
The mean of N n satisfies: E ( N n ) ∼ C (1 /e ) and N n − converges in law to a Poisson distribution of parameter C (1 /e ) (see (7) ).Proof. We can start from (1) and apply singularity analysis. Or we can apply a ready-to-useresult of Bell et al . [4, Thm. 2], which uses the facts (proved in Proposition 6) that the sequences nc n − /c n and c n /a n converge. Proposition 8 ( Size of components — when trees dominate).
The random variable n − S n converges to a discrete limit distribution X given by P ( X = k ) = 1 A (1 /e ) a k e − k k ! , where a k and A ( z ) are given in Proposition . The same holds for n − L n .Proof. The two ingredients used in the proof of Proposition 5 to establish the limit law of n − S n (namely, smoothness of a n /n ! and convergence of c n /a n ), still hold here (see Proposition 6).Lemma 4 gives then the law of n − L n .We now present a collection of classes dominated by trees. Proposition 9.
Let k ≥ . Let A be a decomposable class of graphs that includes all trees, andsuch that all graphs of A avoid the k -spoon (shown in Figure ). Then A is dominated by trees,and the results of Propositions , and hold.Proof. Clearly, it suffices to prove this proposition when A is exactly the class of graphs avoidingthe k -spoon, which we henceforth assume.We partition the set C of connected graphs of A into three subsets: the set C of trees, countedby C = T − T / with T ≡ T ( z ) , the set C of unicyclic graphs (counted by C ), and finally theset C containing graphs with at least two cycles (counted by C ). Hence C = T − T / C + C .We will prove that C has radius of convergence (strictly) larger than /e , and that C is entire.A unicyclic graph belongs to C if and only if all trees attached to its unique cycle have heightless than k . The generating function of cycles is given by: Cyc( z ) = 12 (cid:88) n ≥ z n n = 12 (cid:18) log 11 − z − z − z (cid:19) . (8) SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 9 v v (cid:96) v j v i P v j Pv (cid:96) v v i Qv v (cid:96) Pv v (cid:96) Pv b v a ¯ R (a)(c) (b)(d) ¯ Qv a +1 Q ¯ P ¯ Q ¯ P ¯ P Figure 3.
A core having several cycles and avoiding the k -spoon cannot containa path of length k − .Hence, the basic rules of the symbolic method of [19, Chap. II] give: C ( z ) = Cyc( T k ) = 12 (cid:18) log 11 − T k ( z ) − T k ( z ) − T k ( z ) (cid:19) , (9)where T k counts rooted trees of height less than k and is given in Example 2. Recall from thisexample that T ( z ) equals 1 at its unique dominant singularity /e . Also, T k ( z ) < T ( z ) for all z ∈ [0 , /e ] since T k counts fewer trees than T . In particular, T k (1 /e ) < and C ( z ) has radiusof convergence larger than /e .We now want to prove that C is entire. The (2) -core of a connected graph H is the (possiblyempty) unique maximal subgraph of minimum degree 2. It can be obtained from H by deletingrecursively all vertices of degree 0 or 1 (or, in a non-recursive fashion, all dangling trees of H ).By extension, we call core any connected graph of minimum degree 2. Let ¯ C denote the set ofcores having several cycles and avoiding the k -spoon, and ¯ C the associated generating function.The inequality C ( z ) ≤ ¯ C ( T k ( z )) holds, coefficient by coefficient, because the core of a graph of C has several cycles and avoidsthe k -spoon. Since T k is entire, it suffices to prove that ¯ C is entire. It follows from [6, Thm. 3.1]that it suffices to prove that no graph G of ¯ C contains a path of length k − . So let P =( v , v , . . . , v (cid:96) ) be a path of maximal length in G , and assume that (cid:96) ≥ k − . We will provethat G contains the k -spoon as a minor. Since P is maximal and G is a core, there exist v i and v j , with i ≥ and j ≤ (cid:96) − , such that the edges { v , v i } and { v j , v (cid:96) } belong to G .If i = (cid:96) or j = 0 , let ¯ P be the cycle of G formed of P and the edge { v , v (cid:96) } . Let ¯ Q be anothercycle of G . If ¯ Q contains at most one vertex of P (Figure 3(a)), we find an (cid:96) -spoon by deletingone edge of ¯ P , contracting ¯ Q into a 3-cycle and one of the paths joining P to ¯ Q into a point. If ¯ Q contains at least two vertices v a and v b of P , with a < b (Figure 3(b)), we may assume that ¯ Q consists of the edges { v a , v a +1 } , . . . , { v b − , v b } and of a path Q that only meets P at v a and v b .Let ¯ R denote the cycle formed of the path Q and the path ( v b , v b +1 , . . . , v (cid:96) , v , . . . , v a } . Thenwe obtain a p -spoon, with p ≥ (cid:100) k/ (cid:101) − ≥ k , by contracting the shortest of the cycles ¯ Q and ¯ R into a 3-cycle and deleting an edge ending at v a from the other.Assume now that i < (cid:96) and j > . Suppose first that i ≤ j (Figure 3(c)). By symmetry,we may assume that the cycle ¯ P = ( v , . . . , v i ) is shorter than (or equal in length to) the cycle ¯ P = ( v j , . . . , v (cid:96) ) . In particular, i ≤ (cid:96)/ . Contract ¯ P into a 3-cycle, and remove the edge { v j , v (cid:96) } from ¯ P : this gives a p -spoon with p = (cid:96) − i ≥ (cid:100) (cid:96)/ (cid:101) ≥ k . Assume now that j < i (Figure 3(d)).Consider the three following paths joining v i and v j : ( v i , v i − , . . . , v j ) , ( v i , v , v , . . . , v j ) and ( v i , v i +1 , . . . , v (cid:96) , v j ) . Since the sum of the lengths of these paths is (cid:96) + 2 ≥ k + 1 , one of them, say ( v i , v , v , . . . , v j ) , has length at least k + 1 . That is, j ≥ k . Delete from this path theedge { v i , v } , and contract the cycle formed by the other two paths into a 3-cycle: this gives a j -spoon, with j ≥ k .The simplest non-trivial class of graphs satisfying the conditions of Proposition 9 consists ofgraphs avoiding the 1-spoon. By specializing to k = 1 the proof of that proposition, we find C = Cyc( z ) and C = 0 (since no core having several cycles avoids the 2-path). Hence C ( z ) = T ( z ) − T ( z ) (cid:18) log 11 − z − z − z (cid:19) . More generally, consider the class A ( k ) of graphs avoiding the k -spoon, but also the diamondand the bowtie (both shown in Figure 1): excluding the latter two graphs means that no graphof C can have several cycles, so that C = 0 . Hence the proof of Proposition 9 immediately givesthe following result. Proposition 10 ( No diamond, bowtie or k -spoon). Let k ≥ . Let T ( z ) be the generatingfunction of rooted trees, given by T ( z ) = ze T ( z ) , and let T k ( z ) be the generating function ofrooted trees of height less than k , given in Example 2.Let A ( k ) be the class of graphs avoiding the diamond, the bowtie and the k -spoon. The gener-ating function of connected graphs of A ( k ) is C ( k ) ( z ) = T ( z ) − T ( z ) D ( k ) ( z ) where D ( k ) ( z ) = 12 (cid:18) log 11 − T k ( z ) − T k ( z ) − T k ( z ) (cid:19) . The class A ( k ) is dominated by trees, and the results of Propositions , and hold. In particular,the probability that a random graph of A ( k ) n is connected tends to e − C ( k ) (1 /e ) as n → ∞ . Since T k (1 /e ) tends to T (1 /e ) = 1 as k increases, this limit probability tends to . Figure 4.
A random graph of size n = 541 avoiding the diamond, the bowtieand the 20-spoon. SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 11
A random graph of A ( k ) n is shown in Figure 4 for k = 20 and n = 541 . We have also determinedthe generating function of graphs that avoid the 2-spoon. Proposition 11 ( No 2-spoon).
Let T ( z ) be the generating function of rooted trees, given by T ( z ) = ze T ( z ) . The generating function of connected graphs avoiding the -spoon is C ( z ) = T ( z ) − T ( z ) D ( z ) where D ( z ) = 12 (cid:18) log 11 − ze z − ze z − z e z (cid:19) + z
4! + z e z (cid:18) e z − − z − z (cid:19) . The class of graphs avoiding the -spoon is dominated by trees, and the results of Propositions , and apply.Proof. We first follow the proof of Proposition 9: we write C = T − T / C + C , where C is given by (9) with T k = T = ze z , and C counts connected graphs having several cycles andavoiding the -spoon. Note that C is the first term in the above expression of D ( z ) . Let us nowfocus on C .In Section 10 below, we study the class of graphs that avoid the bowtie, and in particulardescribe the cores of this class (Proposition 31). Since the bowtie contains the 2-spoon as aminor, graphs that avoid the 2-spoon avoid the bowtie as well. Hence we will first determinewhich cores of Proposition 31 have several cycles and avoid the 2-spoon, and then check whichof their vertices can be replaced by a small tree (that is, a tree of height 1) without creating a2-spoon.Clearly, the cores of Proposition 31 that have several cycles are those of Figures 16, 17 and 18.Among the cores of Figure 16, only K avoids the 2-spoon. Moreover, none of its vertices can bereplaced by a non-trivial tree. This gives the term z / in D ( z ) . Among the cores of Figures 17and 18, only the ones drawn on the left-hand sides avoid the 2-spoon. In these cores, only thetwo vertices of degree at least 3 can be replaced by a small tree. The resulting graphs are shownin Figure 5 and give together the contribution
12 ( ze z ) ( e z − − z ) + 12 ( ze z ) (cid:18) e z − − z − z (cid:19) (again an application of the symbolic method of [19, Chap. II]). The proposition follows. . . . at least two vertices . . . at least three vertices Figure 5.
Graphs with several cycles avoiding the -spoon.5. Excluding the diamond and the bowtie: a logarithmic singularity
Let A be the class of graphs avoiding the diamond and the bowtie (both shown in Figure 1).These are the graphs whose components have at most one cycle (Figure 6). They were studied along time ago by Rényi [31] and Wright [34], and the following result has now become a routineexercise. Figure 6.
A random graph of size n = 859 avoiding the diamond and the bowtie. Proposition 12 ( The number of graphs avoiding a diamond and a bowtie).
Let T ( z ) be the generating function of rooted trees, defined by T ( z ) = ze T ( z ) . The generating function ofconnected graphs of A is C ( z ) = T ( z )2 − T ( z ) − T ( z ) . The generating function of graphs of A is A ( z ) = e C ( z ) . As n → ∞ , c n ∼ n ! e n n and a n ∼ n ! 1(2 e ) / Γ(1 / e n n / . (10) In particular, the probability that G n is connected tends to at speed n − / as n → ∞ .Proof. The expression of C ( z ) is obtained by taking the limit k → ∞ in Proposition 10.We now estimate c n and a n via singularity analysis [19, Sect. VI.4]. Recall from Example 2that T ( z ) has a unique dominant singularity, at z = 1 /e , with a singular expansion (6) valid ina ∆ -domain. Thus /e is also the unique dominant singularity of C ( z ) and A ( z ) , and we have,in a ∆ -domain, C ( z ) ∼
14 log (cid:18) − ze (cid:19) and A ( z ) ∼ e ) / (1 − ez ) / . (11)The asymptotic estimates of c n and a n follow. Proposition 13 ( Number of components — no bowtie nor diamond).
The mean andvariance of N n satisfy: E ( N n ) ∼ log n , V ( N n ) ∼ log n , and the random variable N n − log n/ √ log n/ converges in law to a standard normal distribution.Proof. Using (6), the estimate (11) can be refined into C ( z ) = 14 log (cid:18) − ze (cid:19) + λ + O ( √ − ze ) , (12) SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 13 where λ is a constant, and the proposition is a direct application of [19, Prop. IX.14, p. 670].The number of connected components is about / n . However, the size of the root com-ponent is found to be of order n . More precisely, we have the following result. Proposition 14 ( Size of the root component — no bowtie nor diamond).
The normal-ized variable S n /n converges in distribution to a beta law of parameters α = 1 , β = 1 / , withdensity (1 − x ) − / / on [0 , . In fact, a local limit law holds: for x ∈ (0 , and k = (cid:98) xn (cid:99) , n P ( S n = k ) →
14 (1 − x ) − / . The convergence of moments holds as well: for i ≥ , E ( S in ) ∼ Γ(5 / i !Γ( i + 5 / n i . Proof.
Recall that the existence of a local limit law implies the existence of a global one [9,Thm. 3.3]. Thus it suffices to prove the local limit law. But this is easy, starting from therightmost expression in (2), and using (10).For the moments, let us start from (4). Our first task is to obtain an estimate of C ( i +1) ( z ) near /e . Combining (12) and [19, Thm. VI.8, p. 419] gives, for i ≥ , C ( i +1) ( z ) ∼ i !4 (cid:18) e − ze (cid:19) i +1 . We multiply this by the estimate (11) of A ( z ) , apply singularity analysis, and finally use (10)to obtain the asymptotic behaviour of the i th moment of S n . Since these moments characterizethe above beta distribution, we conclude [19, Thm. C.2] that S n /n converges in law to thisdistribution.
20 40 60 80 100 m
10 20 30 40 50 m m Figure 7.
The distribution function P ( L n < m ) for n = 100 . The first plotshows the change of regime at m = n/ , the second the change at m = n/ , thethird the change at m = n/ .We conclude with the law of the size of the largest component, which we derive from gen-eral results dealing with components of logarithmic structures [2]. The following proposition isillustrated by Figure 7. Proposition 15 ( Size of the largest component — no bowtie nor diamond).
The nor-malized variable L n /n converges in law to the first component of a Poisson-Dirichlet distributionof parameter / : for x ∈ (0 , , P ( L n < xn ) → ρ (1 /x ) , where ρ : R + → [0 , is the unique continuous function such that ρ ( x ) = 1 for x ∈ [0 , and for x > , x / ρ (cid:48) ( x ) + 14 ( x − − / ρ ( x −
1) = 0 . The function ρ is infinitely differentiable, except at integer points.A local limit law also holds: for x ∈ (0 , and /x (cid:54)∈ N , n P ( L n = (cid:98) xn (cid:99) ) → (1 − x ) − / x ρ (cid:18) − xx (cid:19) . Proof.
A decomposable class of graphs A is an assembly in the sense of [2, Sec. 2.2]. In particular,it satisfies the conditioning relation [2, Eq. (3.1)]: conditionally to the total size being n , thenumbers C ( n ) i that count connected components of size i , for ≤ i ≤ n , are independent. When A is the class of graphs avoiding the diamond and the bowtie, the estimate (10) of c n tells usthat this assembly is logarithmic in the sense of [2, Eq. (2.15)]: indeed, [2, Eq. (2.16)] holds with m i = c i , y = e and θ = 1 / . Our random variable L n coincides with the random variable L ( n )1 of [2]. We then apply Theorem 6.12 and Theorem 6.8 of [2]: this gives the convergence in law of L n and the local limit law. The distribution function of the limit law is given by [2, Eq. (5.29)],and the differential equation satisfied by ρ follows from [2, Eq. (4.23)]. Remark.
If we push further the singular expansion (12) of C ( z ) , we find a subdominant termin √ − ze , but its influence is never felt in the asymptotics results. We would obtain the sameresults (with possibly different constants) for any C ( z ) having a purely logarithmic singularity.6. Hayman admissibility and extensions
Our next examples (Sections 7 to 10) deal with examples where C ( z ) diverges at ρ with analgebraic singularity. This results in A ( z ) diverging rapidly at ρ . We then estimate a n using thesaddle point method — more precisely, with a black box that applies to Hayman-admissible (or
H-admissible ) functions. Let us first recall what this black box does [19, Thm. VIII.4, p. 565].
Theorem 16.
Let A ( z ) be a power series with real coefficients and radius of convergence ρ ∈ (0 , ∞ ] . Assume that A ( r ) is positive for r ∈ ( R, ρ ) , for some R ∈ (0 , ρ ) . Let a ( r ) = r A (cid:48) ( r ) A ( r ) and b ( r ) = r A (cid:48) ( r ) A ( r ) + r A (cid:48)(cid:48) ( r ) A ( r ) − r (cid:18) A (cid:48) ( r ) A ( r ) (cid:19) . Assume that the following three properties hold: H [Capture condition] lim r → ρ a ( r ) = lim r → ρ b ( r ) = + ∞ . H [Locality condition] For some function θ ( r ) defined over ( R, ρ ) , and satisfying <θ ( r ) < π , one has, as r → ρ , sup | θ |≤ θ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ ) A ( r ) e − iθa ( r )+ θ b ( r ) / − (cid:12)(cid:12)(cid:12)(cid:12) → . H [Decay condition] As r → ρ , sup | θ |∈ [ θ ( r ) ,π ) (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ ) A ( r ) (cid:112) b ( r ) (cid:12)(cid:12)(cid:12)(cid:12) → . We say that A ( z ) is Hayman admissible.
Then the n th coefficient of A ( z ) satisfies, as n → ∞ , [ z n ] A ( z ) ∼ A ( ζ ) ζ n (cid:112) πb ( ζ ) (13) where ζ ≡ ζ n is the unique solution in ( R, ρ ) of the saddle point equation ζA (cid:48) ( ζ ) = nA ( ζ ) . SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 15
Conditions H and H are usually stated in terms of uniform equivalence as r → ρ , but wefind the above formulation more explicit.The set of H-admissible series has several useful closure properties [19, Thm. VIII.5, p. 568].Here is one that we were not able to find in the literature. Theorem 17.
Let A ( z ) = F ( z ) G ( z ) where F ( z ) and G ( z ) are power series with real coefficientsand radii of convergence < ρ F < ρ G ≤ ∞ . Assume that F ( z ) has non-negative coefficients andis Hayman-admissible, and that G ( ρ F ) > . Then A ( z ) is Hayman-admissible.Proof. Let us first prove that the radius of convergence ρ of A ( z ) is ρ F . Clearly, ρ ≥ ρ F . Now,suppose ρ > ρ F . Then A ( z ) is analytic at ρ F . Together with G ( ρ F ) > this implies that F ( z ) = A ( z ) /G ( z ) has an analytic continuation at ρ F , which is impossible by Pringsheim’sTheorem (since F ( z ) has non-negative coefficients) [19, Thm. IV.6, p. 240]. Note also that A ( r ) is positive on an interval of the form [ R, ρ ) (by continuity of G ). Let us now check the threeconditions of Theorem 16. We have a ( r ) = a F ( r ) + a G ( r ) , b ( r ) = b F ( r ) + b G ( r ) , with a F ( r ) = r F (cid:48) ( r ) F ( r ) and b F ( r ) = r F (cid:48) ( r ) F ( r ) + r F (cid:48)(cid:48) ( r ) F ( r ) − r (cid:18) F (cid:48) ( r ) F ( r ) (cid:19) , and similarly for a G and b G . H The capture condition holds for A since it holds for F , given that G ( ρ ) > and ρ G > ρ . H Choose θ ( r ) = θ F ( r ) where θ F ( r ) is a function for which F ( z ) satisfies H and H . Wehave A ( re iθ ) A ( r ) · e − ia ( r ) θ + θ b ( r ) / = F ( re iθ ) F ( r ) e − ia F ( r ) θ + θ b F ( r ) / G ( re iθ ) G ( r ) · e − ia G ( r ) θ + θ b G ( r ) / . (14)By assumption, F satisfies the locality condition: hence F ( re iθ ) F ( r ) · e − ia F ( r ) θ + b F ( r ) θ / = 1 + M ( r, θ ) (15)where sup | θ |≤ θ ( r ) | M ( r, θ ) | → (16)as r → ρ . For r ∈ [ R, ρ ) and | θ | ≤ θ ( r ) , let us expand log G ( re iθ ) in powers of θ : log G ( re iθ ) = log G ( r ) + iθa G ( r ) − θ b G ( r ) + θ S ( r, θ ) where S ( r, θ ) is bounded uniformly in a neighborhood of ( ρ, . We can assume that θ ( r ) → as r → ρ (see [22, Eq. (12.1)]). Thus G ( re iθ ) G ( r ) · e − ia G ( r ) θ + θ b G ( r ) / = e θ S ( r,θ ) = 1 + N ( r, θ ) (17)where sup | θ |≤ θ ( r ) | N ( r, θ ) | → (18)as r → ρ . Putting together Eqs. (14) to (18), we obtain that A ( z ) satisfies H . H We have: (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ ) A ( r ) (cid:112) b ( r ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) F ( re iθ ) G ( re iθ ) F ( r ) G ( r ) (cid:112) b F ( r ) + b G ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) F ( re iθ ) F ( r ) (cid:112) b F ( r ) · G ( re iθ ) G ( r ) (cid:12)(cid:12)(cid:12)(cid:12) for r close to ρ, because b F ( r ) → ∞ as r → ρ while b G ( r ) is bounded around ρ . Also, since G has radius largerthan ρ and G ( ρ ) > , the term G ( re iθ ) /G ( r ) is uniformly bounded in a neighborhood of the circle of radius ρ . Since by assumption, F ( z ) satisfies H , this shows that A ( z ) satisfies it aswell.We will also need a uniform version of Hayman-admissibility for series of the form e uC ( z ) . Theorem 18.
Let C ( z ) be a power series with non-negative coefficients and radius of conver-gence ρ . Assume that A ( z ) = e C ( z ) has radius ρ and is Hayman-admissible. Define b ( r ) = rC (cid:48) ( r ) + r C (cid:48)(cid:48) ( r ) and V ( r ) = C ( r ) − ( rC (cid:48) ( r )) rC (cid:48) ( r ) + r C (cid:48)(cid:48) ( r ) . Assume that, as r → ρ , V ( r ) → + ∞ , (19) C ( r ) V ( r ) / → , (20) b ( r ) / √ V ( r ) = O (1) . (21) Then A ( z, u ) := e uC ( z ) satisfies Conditions (1)–(6), (8) and (9) of [16, Def. 1] . If N n is asequence of random variables such that P ( N n = i ) = [ z n ] C ( z ) i i ![ z n ] e C ( z ) , then the mean and variance of N n satisfy: E ( N n ) ∼ C ( ζ n ) , V ( N n ) ∼ V ( ζ n ) , (22) where ζ n ≡ ζ is the unique solution in (0 , ρ ) of the saddle point equation ζC (cid:48) ( ζ ) = n . Moreover,the normalized version of N n converges in law to a standard normal distribution: N n − E ( N n ) (cid:112) V ( N n ) → N (0 , . Remark.
The set of series covered by this theorem seem to have only a small intersection withthe set of series (of the form g ( z ) F ( uf ( z )) ) covered by Section 4 of [17]. Proof.
With the notation of [16, Def. 1], we have a ( r, u ) = c ( r, u ) = ruC (cid:48) ( r ) = ua ( r ) , b ( r, u ) = ruC (cid:48) ( r ) + r uC (cid:48)(cid:48) ( r ) = ub ( r ) , ¯ a ( r, u ) = ¯ b ( r, u ) = uC ( r ) , ε ( r ) = K (cid:112) V ( r ) (23)for a fixed constant K . Condition (1) of [16, Def. 1] holds for R = ρ , any ζ > and any R ∈ [0 , ρ ) : indeed, the series A ( z, u ) is analytic for | z | < ρ and u ∈ C , and A ( z, is positiveon [0 , ρ ) . Conditions (8) and (9) are nothing but our assumptions (19) and (20). Condition(4) is that b ( r ) → + ∞ as r → ρ : this holds because A is Hayman admissible. Condition(5) requires that b ( r, u ) ∼ b ( r, for r → ρ , uniformly for u ∈ [1 − ε ( r ) , ε ( r )] : this holdsbecause b ( r, u ) /b ( r,
1) = u and ε ( r ) → as r → ρ . Condition (6) requires that a ( r, u ) = a ( r,
1) + c ( r, u −
1) + O ( c ( r, u − )) uniformly for r ∈ (0 , ρ ) and u ∈ [1 − ε ( r ) , ε ( r )] .Since a ( r, u ) = a ( r,
1) + c ( r, u − , this condition obviously holds.We are thus left with Conditions (2) and (3), which are uniform versions (in u ) of the localityand decay conditions H and H defining Hayman admissibility. They can be stated as follows. H (cid:48) [Uniform locality condition] There exists R ∈ (0 , ρ ) such that for any K > , there existsa function δ ( r ) defined over ( R, ρ ) , and satisfying < δ ( r ) < π , such that, as r → ρ , sup | θ |≤ δ ( r ) , | u − |≤ ε ( r ) (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ , u ) A ( r, u ) e − iθa ( r,u )+ θ b ( r,u ) / − (cid:12)(cid:12)(cid:12)(cid:12) → . SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 17 H (cid:48) [Uniform decay condition] As r → ρ , sup | θ |∈ [ δ ( r ) ,π ) | u − |≤ ε ( r ) (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ , u ) A ( r, u ) (cid:112) b ( r, u ) (cid:12)(cid:12)(cid:12)(cid:12) → . We begin with H (cid:48) . Since A ( z ) is H-admissible, let θ ( r ) be a function for which H (and H )holds: A ( re iθ ) A ( r ) e − iθa ( r )+ θ b ( r ) / = 1 + M ( r, θ ) where M ( r ) := sup | θ |≤ θ ( r ) | M ( r, θ ) | tends to 0 as r → ρ . Then, for u ∈ [1 − ε ( r ) , ε ( r )] , A ( re iθ , u ) A ( r, u ) e − iθa ( r,u )+ θ b ( r,u ) / = exp (cid:0) u (cid:0) C ( re iθ ) − C ( r ) − iθa ( r ) + θ b ( r ) / (cid:1)(cid:1) = (1 + M ( r, θ )) u where we have taken the principal determination of log to define (1 + M ( r, θ )) u = exp( u log(1 + M ( r, θ ))) (because M ( r, θ ) is close to ). Thus sup | θ |≤ θ r ) | u − |≤ ε ( r ) (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ , u ) A ( r, u ) e − iθa ( r,u )+ θ b ( r,u ) / − (cid:12)(cid:12)(cid:12)(cid:12) = sup | θ |≤ θ r ) | u − |≤ ε ( r ) | (1 + M ( r, θ )) u − |≤ (1 + ε ( r )) M ( r ) + O (cid:0) (1 + ε ( r )) M ( r ) (cid:1) , and this upper bound tends to 0 as r → ρ . This proves H (cid:48) with δ ( r ) = θ ( r ) .We finally address H (cid:48) . Since A ( z ) satisfies the decay condition H , the quantity N ( r, θ ) := A ( re iθ ) A ( r ) (cid:112) b ( r ) satisfies sup | θ |∈ [ θ ( r ) ,π ] | N ( r, θ ) | → (24)as r → ρ . We have, for u ∈ [1 − ε ( r ) , ε ( r )] , (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ , u ) A ( r, u ) (cid:112) b ( r, u ) (cid:12)(cid:12)(cid:12)(cid:12) = | N ( r, θ ) | u (cid:112) b ( r ) − u √ u ≤ | N ( r, θ ) | ε ( r ) (cid:112) b ( r ) ε ( r ) (cid:112) ε ( r ) , and this tends to uniformly for | θ | ∈ [ θ ( r ) , π ] thanks to (24), (23), (19) and (21).As explained in [16] just below Theorem 2, these eight conditions give the estimates (22) of E ( N n ) and V ( N n ) and imply the existence of a gaussian limit law.We finish this section with a simple but useful result of products of series [5, Thm. 2]. Proposition 19.
Let F ( z ) = (cid:80) n f n z n and G ( z ) = (cid:80) n g n z n be power series with radii ofconvergence ≤ ρ F < ρ G ≤ ∞ , respectively. Suppose that G ( ρ F ) (cid:54) = 0 and the sequence ( f n ) n ≥ is smooth. Then [ z n ] F ( z ) G ( z ) ∼ G ( ρ F ) f n . Graphs with bounded components: C ( z ) is a polynomial Let C be a finite class of connected graphs, and let A be the class of graphs with connectedcomponents in C . Note that A is minor-closed if and only if C itself is minor-closed. This is thecase for instance if C is the class of graphs of size at most k . In general, we denote by k the sizeof the largest graphs of C . We assume that C ( z ) is aperiodic.We begin with the enumeration of the graphs of A . The following proposition is a bit moreprecise than the standard result on exponentials of polynomials [19, Cor. VIII.2, p. 568], sinceit makes explicit the behaviour of the term b ( ζ ) occurring in the saddle point estimate (13). Proposition 20 ( The number of graphs with small components).
Write the generatingfunction of graphs of C as C ( z ) = k (cid:88) i =0 c i i ! z i . (25) The generating function of graphs of A is A ( z ) = e C ( z ) . As n → ∞ , a n ∼ n ! 1 √ πkn A ( ζ ) ζ n (26) where ζ ≡ ζ n is defined by ζC (cid:48) ( ζ ) = n and satisfies ζ = αn /k + β + O ( n − /k ) (27) with α = (cid:18) ( k − c k (cid:19) /k and β = − ( k − c k − kc k . (28) The probability that G n is connected is of course zero as soon as n > k .Proof. The series A ( z ) is H-admissible ([19, Thm. VIII.5, p. 568]) and Theorem 16 applies. Thesaddle point equation ζC (cid:48) ( ζ ) = n is an irreducible bivariate polynomial in ζ and n , of degree k in ζ . Consider /n as a small parameter x . By [33, Prop. 6.1.6], the saddle point ζ admits anexpansion of the form ζ = (cid:88) i ≥ i α i n − i/k , (29)for some integer i and complex coefficients α i . Using Newton’s polygon method [19, p. 499],one easily finds i = − and the values (28) of the first two coefficients.Since b ( r ) = rC (cid:48) ( r ) + r C (cid:48)(cid:48) ( r ) has leading term kc k r k / ( k − , the first order expansion of b ( ζ ) reads b ( ζ ) = kn + O ( n ( k − /k ) , and the asymptotic behaviour of a n follows.Again, the following proposition is more precise than the statement found, for instance, in [11,Thm. I], because our estimates of E ( N n ) and V ( N n ) are explicit. Note in particular that E ( N n ) ∼ n/k suggests that most components have maximal size k . Proposition 21 ( Number of components — Graphs with small components).
Assumethat the coefficient c k − in (25) is non-zero. The mean and variance of N n satisfy: E ( N n ) ∼ nk , V ( N n ) ∼ c k − k · k ! α k − n ( k − /k , where α is given by (28) , and the random variable N n − E ( N n ) √ V ( N n ) converges in law to a standardnormal distribution. SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 19
Proof.
We apply [11, Thm. I] (we can also apply Theorem 18 if k > ). Still denoting the saddlepoint by ζ ≡ ζ n , we just have to find estimates of µ n = C ( ζ ) and σ n = C ( ζ ) − ( ζC (cid:48) ( ζ )) ζC (cid:48) ( ζ ) + ζ C (cid:48)(cid:48) ( ζ ) . Given (27), we obtain µ n = nk + c k − k ! α k − n ( k − /k + O ( n ( k − /k ) ,ζ C (cid:48)(cid:48) ( ζ ) = ( k − n − c k − ( k − α k − n ( k − /k + O ( n ( k − /k ) , and finally σ n = µ n − n n + ζ C (cid:48)(cid:48) ( ζ ) = c k − k · k ! α k − n ( k − /k + O ( n ( k − /k ) . Since there are approximately n/k components, one expects the size S n of the root componentto be k . This is indeed the case, as illustrated in Figure 8. Proposition 22 ( Size of the components — Graphs with small components).
Thedistribution of S n converges to a Dirac law at k : P ( S n = j ) → (cid:40) if j = k , otherwise.The same holds for the size L n of the largest component.Proof. We combine the second formulation in (2) with the estimate (26) of a n . This gives P ( S n = j ) ∼ n c j ( j − A ( ζ n − j ) A ( ζ n ) ζ nn ζ n − jn − j . Figure 8.
A random graph of size n = 1171 with component size at most 3.Observe that most components have size 3, so that the root component is verylikely to have size 3. Clearly, it suffices to prove that this probability tends to if j < k . So let us assume j < k .Since ζ n is increasing with n , it suffices to prove that n ζ nn ζ n − jn − j → . (30)Recall from (27) and (29) that ζ n admits an expansion of the form ζ n = αn /k + k − (cid:88) i =0 n − i/k β i + O (1 /n ) . This gives, for some constants γ i , n log ζ n = nk log n + n log α + k (cid:88) i =1 γ i n − i/k + O ( n − /k ) . Hence ( n − j ) log ζ n − j = n − jk log n + ( n − j ) log α + k (cid:88) i =1 γ i n − i/k + O (1) . This gives n log ζ n − ( n − j ) log ζ n − j − log n = j − kk log n + O (1) , and (30) follows, since j < k . Since L n ≥ S n , the behaviour of L n is then clear.8. Forests of paths or caterpillars: a simple pole in C ( z ) Let A be a decomposable class (for instance defined by excluding connected minors), withgenerating function A ( z ) = exp( C ( z )) . Assume that C ( z ) = α − z/ρ + D ( z ) , (31)where D has radius of convergence larger than ρ . Of course, we assume α > . Proposition 23 ( The number of graphs — when C has a simple pole). Assume thatthe above conditions hold, and let β = D ( ρ ) . As n → ∞ , c n ∼ n ! αρ − n and a n ∼ n ! α / e α/ β √ πn / ρ − n e √ αn . (32) In particular, the probability that G n is connected tends to at speed n / e − √ αn .Proof. The asymptotic behaviour of c n follows from [19, Thm. IV.10, p. 258]. To obtain theasymptotic behaviour of a n , we first write A ( z ) = F ( z ) G ( z ) with F ( z ) = exp (cid:18) α − z/ρ (cid:19) and G ( z ) = e D ( z ) , (33)where G ( z ) has radius of convergence larger than ρ . To estimate the coefficients of F , we applythe ready-to-use results of Macintyre and Wilson [25, Eqs. (10)–(14)], according to which, for α, γ > and a non-negative integer k , [ z n ] (cid:18) log 11 − z (cid:19) k − z ) γ exp (cid:18) α − z (cid:19) ∼ α / e α/ √ πn / (cid:16) nα (cid:17) γ/ (cid:18) log n (cid:19) k e √ αn . (34)This gives f n := [ z n ] F ( z ) ∼ α / e α/ √ πn / ρ − n e √ αn . This shows in particular that f n − /f n tends to ρ as n → ∞ , so that we can apply Proposition 19to (33) and conclude. SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 21
Proposition 24 ( Number of components — when C has a simple pole). Assume (31) holds. The mean and variance of N n satisfy: E ( N n ) ∼ √ αn, V ( N n ) ∼ √ αn/ , and the random variable N n −√ αn ( αn/ / converges in law to a standard normal distribution.Proof. We apply Theorem 18. The H-admissibility of A ( z ) follows from Theorem 17, using (33)and the H-admissibility of exp( α/ (1 − z/ρ )) (see [19, p. 562]). Conditions (19)–(21) are thenreadily checked, using C ( r ) ∼ α − r/ρ , b ( r ) ∼ α (1 − r/ρ ) and V ( r ) ∼ α − r/ρ ) . We thus conclude that the normalized version of N n converges in law to a standard normaldistribution. For the asymptotic estimates of E ( N n ) and V ( N n ) , we use (22) with the saddlepoint estimate ζ n = ρ − ρ (cid:112) α/n + O (1 /n ) . Since there are approximately √ n components, one may expect the size S n of the root com-ponent to be of the order of √ n . Proposition 25 ( Size of the root component — when C has a simple pole). Thenormalized variable S n / (cid:112) n/α converges in distribution to a Gamma (2 , law of density xe − x on [0 , ∞ ) . In fact, a local limit law holds: for x > and k = (cid:98) x (cid:112) n/α (cid:99) , (cid:112) n/α P ( S n = k ) → xe − x . The convergence of moments holds as well: for i ≥ , E ( S in ) ∼ ( i + 1)!( n/α ) i/ . Proof.
For the local (and hence global) limit law, we simply combine (2) with (32). For themoments, we start from (4), with C ( i +1) ( z ) = α ( i + 1)! ρ i +1 (1 − z/ρ ) i +2 + D ( i +1) ( z ) . Let us first observe that (32) implies that a n /n ! is smooth. We can thus apply Proposition 19to the product D ( i +1) ( z ) A ( z ) , which gives [ z n − i − ] D ( i +1) ( z ) A ( z ) n [ z n ] A ( z ) ∼ D ( i +1) ( ρ ) n a n − i − ( n − i − n ! a n ∼ D ( i +1) ( ρ ) n ρ i → . Figure 9.
A random forest of paths of size n = 636 (left) and a forest ofcaterpillars of size n = 486 (right). We thus have a n ( n − E ( S in ) ∼ [ z n − i − ] α ( i + 1)! ρ i +1 (1 − z/ρ ) i +2 exp (cid:18) α − z/ρ + D ( z ) (cid:19) . (35)Now (34) gives [ z n − i − ] α ( i + 1)! ρ i +1 (1 − z/ρ ) i +2 exp (cid:18) α − z/ρ (cid:19) ∼ α ( i + 1)! α / e α/ √ πn / (cid:16) nα (cid:17) i/ ρ − n e √ αn . (36)In particular, this sequence of coefficients is smooth. Hence by Proposition 19, the asymptoticbehaviour of (35) only differs from (36) by a factor e β , where β = D ( ρ ) . Combined with (32),this gives the limiting i th moment of S n . Since these moments characterize the above Gammadistribution, we can conclude [19, Thm. C.2] that S n / (cid:112) n/α converges in law to this distribution.We now present two classes for which C ( z ) has a simple isolated pole (Figure 9): forests ofpaths, and forests of caterpillars (a caterpillar is a tree made of a simple path to which leavesare attached; see Figure 1). In forests of paths, the excluded minors are the triangle K andthe 3-star. The fact that N n converges in probability to (cid:112) n/ for this class was stated in [26,p. 587]. For forests of caterpillars, the excluded minors are K and the tree shown in Table 1( th line). This class is also considered in [6]. It is also the class of graphs of pathwidth 1. Proposition 26 ( Forests of paths or caterpillars).
The generating functions of paths andof caterpillars are respectively C p ( z ) = z (2 − z )2(1 − z ) and C c ( z ) = z ( e z − − ze z ) + ze z − z . (37) For both series, Condition (31) is satisfied and Propositions
23, 24 and hold. For paths wehave ρ = 1 , α = 1 / and β := D ( ρ ) = 0 . For caterpillars, ρ (cid:39) . is the only real such that ρe ρ = 1 , α = (1 − ρ ) ρ ) (cid:39) . and β = ρ (cid:0)
10 + 3 ρ − ρ − ρ (cid:1) ρ ) (cid:39) . . (38) Proof.
The expression of C p ( z ) is straightforward. One can also write C p ( z ) = 12(1 − z ) + z − , which gives D p (1) = 0 . Let us now focus on caterpillars. Let us call star a tree in which allvertices, except maybe one, have degree 1. By a rooted star we mean a star with a marked vertexof maximum degree: hence the root has degree 0 for a star with 1 vertex, 1 for a star with 2vertices, and at least 2 otherwise. Clearly, there are n rooted stars on n labelled vertices, so thattheir generating function is S • ( z ) = (cid:88) n ≥ z n ( n − ze z . The generating function of (unrooted) stars is S ( z ) = S • ( z ) − z ze z − z (because all stars have only one rooting, except the star on 2 vertices which has two). Now acaterpillar is either a star, or is a (non-oriented) chain of at least two rooted stars, the first andlast having at least 2 vertices each. This gives C ( z ) = S ( z ) + ( S • ( z ) − z ) − S • ( z )) , which is equivalent to the right-hand side of (37). SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 23
The series C c ( z ) is meromorphic on C , with a unique dominant pole at ρ , and its behaviouraround this point is easily found using a local expansion of ze z at ρ : C c ( z ) = α − z/ρ + β + O (1 − z/ρ ) , with α and β as in (38).For forests of paths, we have also obtained the limit law of the size L n of the largest component.It is significantly larger than the root component ( √ n log n instead of √ n ). Proposition 27 ( Size of the largest component — forests of paths).
In forests of paths,the (normalized) size of the largest component converges in law to a Gumbel distribution: for x ∈ R and as n → ∞ , P (cid:32) L n − (cid:112) n/ n (cid:112) n/ < x (cid:33) → exp (cid:18) − e − x/ √ (cid:19) . Proof.
We start from (5), where k = (cid:112) n/ n + x ) (39)and the generating function of paths of size less than k is: C [ k ] ( z ) = z z − z k − z ) . (40)Using a saddle point approach for integrals [19, p. 552], we will find an estimate of [ z n ] exp( C [ k ] ( z )) = 12 iπ (cid:90) C r exp( C [ k ] ( z )) dzz n +1 , (41)where the integration contour is any circle C r of center and radius r < .Let us first introduce some notation: we denote C [ k ] ( z ) by K ( z ) , the integrand in (41) by F ,and its logarithm by f : K ( z ) = C [ k ] ( z ) , F ( z ) = exp( K ( z )) z n +1 , f ( z ) = K ( z ) − ( n + 1) log z. We choose the radius r ≡ r n that satisfies the saddle point equation F (cid:48) ( r ) = 0 , or equivalently f (cid:48) ( r ) = 0 or rK (cid:48) ( r ) = n + 1 . Note that rK (cid:48) ( r ) increases from to ∞ as r grows from to , so that the solution of thisequation is unique and simple to approximate via bootstrapping. We find: r = 1 − √ n + e − x/ √ nn + O (cid:18) n (cid:19) . (42) Gaussian approximation.
Let θ ∈ (0 , π ) . By expanding the function g : θ (cid:55)→ f ( re iθ ) in theneighbourhood of θ = 0 , we find, for | θ | ≤ θ : (cid:12)(cid:12) f ( re iθ ) − f ( r ) + θ r f (cid:48)(cid:48) ( r ) / (cid:12)(cid:12) ≤ θ | α |≤ θ (cid:12)(cid:12)(cid:12) g (3) ( α ) (cid:12)(cid:12)(cid:12) , (43)with (cid:12)(cid:12)(cid:12) g (3) ( α ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) − ire iα f (cid:48) ( re iα ) − ir e iα f (cid:48)(cid:48) ( re iα ) − ir e iα f (cid:48)(cid:48)(cid:48) ( re iα ) (cid:12)(cid:12) ≤ K (cid:48) ( r ) + n + 1 r + 3 K (cid:48)(cid:48) ( r ) + 3 n + 1 r + K (cid:48)(cid:48)(cid:48) ( r ) + 2 n + 1 r . By combining the expression (40) of K ( z ) = C [ k ] ( z ) and the saddle point estimate (42), we findthat K (cid:48) ( r ) = ( n + 1) /r ∼ n , that K (cid:48)(cid:48) ( r ) ∼ √ n / and finally that K (cid:48)(cid:48)(cid:48) ( r ) ∼ n . This termdominates the above bound on | g (3) ( α ) | . Hence, if θ ≡ θ ( n ) = o ( n − / ) , (44) we find, by taking the exponential of (43), F ( re iθ ) ∼ F ( r ) e − θ r f (cid:48)(cid:48) ( r ) / , uniformly in | θ | ≤ θ . Completion of the Gaussian integral.
We split the integral (41) into two parts, dependingon whether | θ | ≤ θ or | θ | ≥ θ . The first part is (cid:90) θ − θ F ( re iθ ) re iθ dθ π ∼ rF ( r )2 π (cid:90) θ − θ e − θ r f (cid:48)(cid:48) ( r ) / e iθ dθ. As argued above, r f (cid:48)(cid:48) ( r ) ∼ K (cid:48)(cid:48) ( r ) ∼ √ n / . Hence, if we choose θ ≡ θ ( n ) such that θ n / → ∞ (which is compatible with (44), for instance if θ = n − / , (45)which we henceforth assume), we obtain (cid:90) θ − θ F ( re iθ ) re iθ dθ π ∼ F ( r )2 π (cid:112) f (cid:48)(cid:48) ( r ) (cid:90) θ r √ f (cid:48)(cid:48) ( r ) − θ r √ f (cid:48)(cid:48) ( r ) e − α / e iα/ ( r √ f (cid:48)(cid:48) ( r )) dα ∼ F ( r )2 π (cid:112) f (cid:48)(cid:48) ( r ) (cid:90) R e − α / dα ∼ F ( r ) (cid:112) πf (cid:48)(cid:48) ( r ) ∼ e / e √ n / √ πn / exp (cid:18) − e − x/ √ (cid:19) (46)by (42). The second part of the integral can be neglected.
The second part of the integral (41) is (cid:90) θ < | θ | <π F ( re iθ ) re iθ dθ π , and we want to prove that it is dominated by (46). It suffices to prove that for θ < | θ | < π , | F ( re iθ ) | = o (cid:32) F ( r ) (cid:112) f (cid:48)(cid:48) ( r ) (cid:33) . (47)Let us denote z = re iθ and z = re iθ . We have | F ( re iθ ) | F ( r ) = | exp( K ( z ) − K ( r )) |≤ exp( | K ( z ) | − K ( r )) = exp (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) z z − z k − z ) (cid:12)(cid:12)(cid:12)(cid:12) − r − r − r k − r ) (cid:19) ≤ exp (cid:18) r + r k | − z | − r − r k − r ) (cid:19) ≤ exp (cid:18) r + r k | − z | − r − r k − r ) (cid:19) = exp (cid:18) − n / √ o (1)) (cid:19) , given the values (39), (42) and (45) of k , r and θ . Since f (cid:48)(cid:48) ( r ) ∼ √ n / , we conclude that (47)holds. Conclusion.
We have now established that the integral (41) is dominated by its first part, andis thus equivalent to (46). To obtain the limiting distribution function, it remains to divide thisestimate by a n /n ! . The asymptotic behaviour of a n is given by (32), with α = 1 / , ρ = 1 and β = 0 , and this concludes the proof. SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 25
Figure 10.
A random graph of size n = 1034 avoiding the 3-star.9. Graphs with maximum degree 2: a simple pole and a logarithmin C ( z ) Let A be the class of graphs of maximum degree 2, or equivalently, the class of graphs avoidingthe 3-star (Figure 10). The connected components of such graphs are paths or cycles. This classdiffers from those studied in the previous section in that the series C ( z ) has now, in additionto a simple pole, a logarithmic singularity at its radius of convergence ρ . As we shall see, thelogarithm changes the asymptotic behaviour of the numbers a n , but the other results remainunaffected. The proofs are very similar to those of the previous section. Proposition 28 ( The number of graphs of maximum degree 2).
The number of connectedgraphs (paths or cycles) of size n in the class A is c n = n ! / n − / for n ≥ (with c = c = 1 ) and the associated generating function is C ( z ) = z (2 − z + z )4(1 − z ) + 12 log 11 − z . The generating function of graphs of A is A ( z ) = e C ( z ) = 1 √ − z exp (cid:18) z (2 − z + z )4(1 − z ) (cid:19) . As n → ∞ , a n ∼ n ! 12 √ eπn / e √ n . In particular, the probability that G n is connected tends to at speed n / e −√ n as n → ∞ .Proof. Again, the exact results are elementary. To obtain the asymptotic behaviour of a n , wewrite A ( z ) = F ( z ) G ( z ) with F ( z ) = 1 √ − z exp (cid:18) − z ) (cid:19) and G ( z ) = exp (cid:18) − − z (cid:19) (48)and combine Proposition 19 with (34).For the number of components, we find the same behaviour as in the case of a simple pole(Proposition 24 with α = 1 / ). We have also determined the expected number of cyclic compo-nents. Proposition 29 ( Number and nature of components — Graphs of maximum degree2).
The mean and variance of N n satisfy: E ( N n ) ∼ (cid:112) n/ , V ( N n ) ∼ (cid:112) n/ , and the random variable N n − √ n/ n/ / converges in law to a standard normal distribution.The expected number of cycles in G n is of order (log n ) / .Proof. We want to apply Theorem 18. To prove that A ( z ) is Hayman-admissible, we applyTheorem 17 to (48). This reduces our task to proving that F ( z ) is H-admissible, which is donealong the same lines as [19, Ex. VIII.7, p. 562] (see also the footnote of [22, p. 92], and Lemma 1in [17]). Conditions (19)–(21) are readily checked. The asymptotic estimates of E ( N n ) and V ( N n ) are obtained through (22), using the saddle point estimate ζ n = 1 − / √ n + O (1 /n ) .The bivariate generating function of graphs of A , counted by the size (variable z ) and thenumber of cycles (variable v ) is ˜ A ( z, v ) = exp (cid:18) z + z − z ) + v Cyc( z ) (cid:19) , where Cyc( z ) is given by (8). By differentiating with respect to v , the expected number of cyclesin G n is found to be: [ z n ] Cyc( z ) A ( z )[ z n ] A ( z ) . The asymptotic behaviour of [ z n ] A ( z ) = a n /n ! has been established in Proposition 28. We deter-mine an estimate of [ z n ] Cyc( z ) A ( z ) in a similar fashion, using a combination of Proposition 19and (34). We find [ z n ] Cyc( z ) A ( z ) ∼ log n √ eπn / e √ n , and the result follows.The size of the root component is still described by Proposition 25, with α = 1 / . The proofis very similar, with now C ( i +1) ( z ) = i !2 2 + i − z (1 − z ) i +2 − i =1 , where i =1 is if i = 1 and is 0 otherwise.10. Excluding the bowtie: a singularity in (1 − z/ρ ) − / We now denote by A the class of graphs avoiding the bowtie (Figure 11). The followingproposition answers a question raised in [27]. Proposition 30 ( The generating function of graphs avoiding a bowtie).
Let T ≡ T ( z ) be the generating function of rooted trees, defined by T ( z ) = ze T ( z ) . The generating function ofconnected graphs in the class A is C ( z ) = T (cid:0) − T + T (cid:1) e T − T + 12 log (cid:18) − T (cid:19) + T (cid:0) − T + 18 T − T − T (cid:1) − T ) . (49) The generating function of graphs of A is A ( z ) = e C ( z ) . This is the most delicate enumeration of the paper. The key point is the following character-ization of cores (graphs of minimum degree 2) avoiding the bowtie.
Proposition 31.
The cores that avoid the bowtie are: • the empty graph, • all cycles, • K , with one edge possibly subdivided, as shown in Figure , • the graphs of Figures and . SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 27
Figure 11.
A random graph of size n = 758 avoiding the bowtie.We will first establish a number of properties of cores avoiding a bowtie. Recall that a chord of a cycle C is an edge, not in C , joining two vertices of C . Lemma 32.
Let C = ( v , v , . . . , v n − ) be a cycle in a core G avoiding the bowtie. Let us write v n = v and v n +1 = v . Every chord of C joins vertices that are at distance on C (we say thatit is a short chord). Moreover, C has at most two chords. If it has two chords, say { v , v } and { v i , v i +2 } , with ≤ i ≤ n − , then v i = v or v i +2 = v .Proof. If a chord were not short, contracting it (together with some edges of C ) would give abowtie. Figure 12 then proves the second statement, which can be loosely restated as follows:the two chords cross and their four endpoints are consecutive on C . yesno no Figure 12.
The relative positions of two (short) chords in a cycle. The firsttwo configurations are not valid, as these graphs contain a bowtie (shown withgray/blue edges).
Lemma 33.
Let C be a cycle of maximal length in a core G avoiding the bowtie. Let v be an external vertex, that is, a vertex not belonging to C . Then v is incident to exactly two edges,both ending on C . The endpoints of these edges are at distance on C .Proof. Since G is a core, v belongs to a cycle C (cid:48) . Since G is connected and avoids the bowtie, C (cid:48) shares at least two vertices with C . Thus let P and P be two vertex-disjoint paths (taken from C (cid:48) ) that start from v and end on C without hitting C before. Let v and v be their respectiveendpoints on C . Then v and v lie at distance at least 2 on C , otherwise C would not havemaximal length. Now contracting the path P P into a single edge gives a chord of C . By the previous lemma, this chord must be short, so that v and v are at distance exactly 2. Since C has maximal length, P and P have length 1 each, and thus are edges.Assume now that v has degree at least 3, and let e be a third edge (distinct from P and P )adjacent to v . Again, e must belong to a cycle, sharing at least two vertices with C , and thesame argument as before shows that e ends on C . But then Figure 13 shows that G contains abowtie. v C no Figure 13.
A cycle C with an external vertex v of degree at least 3. The cycleshown in gray/blue has a chord that is not short, the contraction of which givesa bowtie. Lemma 34.
Let C be a cycle of maximal length in a core G avoiding the bowtie. If C has twochords, it contains all vertices of G .Proof. Let e and e be the two chords of C . Lemma 32 describes their relative positions. Let v be a vertex not in C . Lemma 33 describes how it is connected to C . Contract one of the twoedges incident to v to obtain a chord of C . By Lemma 32, this chord must be a copy of e or e . But then Figure 14(a) shows that G contains a bowtie (delete the two bold edges). no v eC (a) C v (b)
Figure 14.
A cycle C with (a) two chords and an external vertex v (b) onechord and an external vertex v . Lemma 35.
Let C be a cycle of maximal length in a core G avoiding the bowtie. If C has achord e , all external vertices of C are adjacent to the endpoints of e .Proof. Let v be external to C . Contract one of the incident edges. This gives a chord e (cid:48) . If e (cid:48) is a copy of e , then we are done. Otherwise, the relative positions of e and e (cid:48) are describedby Lemma 32. But then Figure 14(b) shows that C has not maximal length (consider the cycleshown with the dotted line). Lemma 36.
Let C be a cycle of maximal length in a core G avoiding the bowtie. If C hasseveral external vertices, they are adjacent to the same points of C .Proof. Consider two external vertices v and v . Lemma 33 describes how each of them isconnected to C . Contract an edge incident to v and an edge incident to v . This gives twochords of C . Either these two chords are copies of one another, which means that v and v areadjacent to the same points of C . Or the relative position of these two chords is as described inLemma 32. But then Figure 15 shows that G contains a bowtie (contract e ). SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 29 no ev v Figure 15.
A cycle C with two external vertices. Proof of Propositions 30 and 31.
Observe that a graph G avoids the bowtie if and only if its core(defined as its maximal subgraph of degree 2) avoids it. Hence, if ¯ C ( z ) denotes the generatingfunction of non-empty cores avoiding the bowtie, we have C ( z ) = T ( z ) − T ( z ) C ( T ( z )) . (50)Using the above lemmas, we can now describe and count non-empty cores avoiding the bowtie.We start with cores reduced to a cycle: their contribution to ¯ C ( z ) is given by (8). We nowconsider cores G having several cycles. Let C be a cycle of G of maximal length, chosen so thatit has the largest possible number of chords. By Lemma 32, this number is 2, 1 or 0.If C has two chords, it contains all vertices of G (Lemma 34). By Lemma 32 and Figure 12(right), either all vertices have degree 3 and G = K , or G consists of K where one edge issubdivided (Figure 16). ≥ Figure 16. K with a subdivided edge.This gives the generating function z
4! + z · · z − z , (51)where, in the second term, we read first the choice of the 4 vertices of degree 3 forming a K ,then the choice of one edge of this K , and finally the choice of a (directed) path placed alongthis edge.Assume now that C has exactly one chord e . By Lemma 35, all external vertices are adjacentto the endpoints of e . To avoid problems with symmetries, we count separately the cores where C has length , or length ≥ (Figure 17). This gives the generating function z e z − − z ) + z e z − z − z . (52)In the second term, the factor z / (1 − z ) accounts for the directed chain of vertices of degree 2lying on the maximal cycle. at least one vertex ≥ at least two vertices . . .e . . .e Figure 17.
A maximal cycle with one chord e and external vertices. Assume finally that C has no chord. By Lemma 36, all external vertices are adjacent to thesame points of C . Again, we treat separately the cases where C has length , or length ≥ (Figure 18). This gives the generating function z e z − − z − z /
2) + z e z − − z ) z − z . (53) ≥ at least two vertices . . . at least three vertices . . . Figure 18.
A maximal cycle C with no chord and several external vertices.Putting together the contributions (8), (51), (52) and (53) gives the value of ¯ C ( z ) (the gen-erating function of cores) from which we obtain the series C ( z ) using (50).We now derive asymptotic results from Proposition 30. Proposition 37 ( The asymptotic number of graphs avoiding the bowtie). As n → ∞ , c n ∼ n ! e − / √ π e n √ n and a n ∼ n ! ( e − / / e / − e/ √ π e n n / exp (cid:18)
32 ( e − / / n / (cid:19) . Proof.
Let us first recall that the series T ( z ) has radius of convergence /e , and can be continuedanalytically on the domain D := C \ [1 /e, + ∞ ) . In fact, T ( z ) = − W ( − z ) , where W is the(principle branch of the) Lambert function [12]. The singular behaviour of T ( z ) near /e is givenby (6). Moreover, the image of D by T avoids the half-line [1 , + ∞ ) .It thus follows from the expression (49) of C ( z ) that C ( z ) and A ( z ) are analytic in the domain D . Moreover, we derive from (6) that, as z approaches /e in a ∆ -domain, C ( z ) ∼ e − / √ √ − ze . (54)The above estimate of c n then follows from singularity analysis.We now embark with the estimation of a n . We first prove (see Proposition 40 in the appendix)that A ( z ) is H-admissible. We then apply Theorem 16. The saddle point equation reads ζC (cid:48) ( ζ ) = n . Using the singular expansion (6) of T ( z ) , and a similar expansion for T (cid:48) ( z ) , this reads e − / √ − ζe ) / + 14(1 − ζe ) + O (cid:18) − ζe ) / (cid:19) = n. (55)This gives the saddle point as ζ = 1 e − ( e − / / en / − en + O ( n − / ) . (56)We now want to obtain estimates of the values A ( ζ ) , ζ n and b ( ζ ) occurring in Theorem 16.Refining (54) into C ( z ) = e − / √ √ − ze + 14 log 12(1 − ze ) + 198 − e O ( √ − ze ) , (57)we find C ( ζ ) = ( e − / / n / + 16 log ne − / − e O ( n − / ) , SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 31 which gives A ( ζ ) ∼ e − e ( e − / / n / exp(( e − / / n / ) . (58)It then follows from (56) that ζ n ∼ e − / exp (cid:16) − n − ( e − / / n / / (cid:17) . (59)Finally, b ( r ) = rC (cid:48) ( r ) + r C (cid:48)(cid:48) ( r ) ∼ √ e − / − re ) / , (60)so that b ( ζ ) ∼ e − / / n / . Putting this estimate together with (58) and (59), we obtain the estimate of a n /n ! given in theproposition. Proposition 38 ( Number of components — no bowtie).
The mean and variance of N n satisfy: E ( N n ) ∼ ( e − / / n / , V ( N n ) ∼
23 ( e − / / n / , and the random variable N n − E ( N n ) √ V ( N n ) converges in law to a standard normal distribution.Proof. We want to apply Theorem 18. By Proposition 40, A ( z ) is H-admissible. Conditions (19)–(21) are readily checked, using C ( z ) ∼ e − / √ √ − ze , b ( r ) ∼ √ e − / − ze ) / and V ( r ) ∼ √ e − / √ − ze . The asymptotic estimates of E ( N n ) and V ( N n ) are obtained through (22), using the saddle pointestimate (56).Since there are approximately n / components, one may expect the size S n of the rootcomponent to be of the order of n / . More precisely, we have the following result. Proposition 39 ( Size of the root component — no bowtie).
The normalized variable ( e − / / S n / (2 n / ) converges in distribution to a Gamma (3 / , law, of density √ xe − x / √ π on [0 , ∞ ) . In fact, a local limit law holds: for x > and k = (cid:98) x n / ( e − / / (cid:99) , n / ( e − / / P ( S n = k ) → (cid:114) xπ e − x . The convergence of moments holds as well: for i ≥ , E ( S in ) ∼ Γ( i + 3 / / (cid:18) n / ( e − / / (cid:19) i . Proof.
The local (and hence global) limit law follows directly from Proposition 37, using (2).For the convergence of the moments, we start from (4). We first prove (see Proposition 40 inthe appendix) that C ( i +1) ( z ) A ( z ) is H-admissible. We then apply Theorem 16 to estimate thecoefficient of z n in this series (we will replace n ny n − i − later). Our calculations mimic thoseof Proposition 37, but the saddle point equation now reads ζC (cid:48) ( ζ ) + ζ C ( i +2) ( ζ ) C ( i +1) ( ζ ) = n, where ζ ≡ ζ ( i ) n depends on i and n . Comparing with the original saddle point equation (55), andusing the estimate (69) of C ( i ) ( z ) , this reads e − / √ − ζe ) / + 7 + 4 i − ζe ) + O (cid:18) − ζe ) / (cid:19) = n. This gives the saddle point as ζ = 1 e − ( e − / / en / − i en + O ( n − / ) . (61)We now want to obtain estimates of C ( i +1) ( ζ ) A ( ζ ) , ζ n and b i ( ζ ) . We first derive from (57) that C ( ζ ) = ( e − / / n / + 16 log ne − / − e − i O ( n − / ) . This gives A ( ζ ) ∼ e − e − i ( e − / / n / exp(( e − / / n / ) . (62)Moreover, we derive from (69) that C ( i +1) ( ζ ) ∼ (2 i + 1)!2 i i ! e i +1 n i/ ( e − / i/ . (63)It then follows from (61) that ζ n ∼ e − / − i/ exp (cid:16) − n − ( e − / / n / / (cid:17) . (64)Finally, (71) and (60) give b i ( ζ ) ∼ b ( ζ ) ∼ e − / / n / . Putting this estimate together with (62), (63) and (64), we obtain [ z n ] C ( i +1) ( z ) A ( z ) ∼ (2 i + 1)!2 i i ! ( e − / / e / − e/ √ π e n + i +1 n / n i/ ( e − / i/ exp (cid:18)
32 ( e − / / n / (cid:19) . We finally replace n by n − i − (the only effect is to replace e n + i +1 by e n ), and divide by theestimate of na n /n ! given in Proposition 37: this gives the estimate of the i th moment of S n asstated in the proposition, and concludes the proof.11. Final comments and further questions
Random generation
For each of the classes A that we have studied, we have designed an associated Boltzmannsampler , which generates a graph G of A with probability P ( G ) = x | G | | G | ! A ( x ) , (65)where x > is a fixed parameter such that A ( x ) converges. We refer to [18, Sec. 4] for generalprinciples on the construction of exponential Boltzmann samplers, and only describe how wehave addressed certain specific difficulties. Most of them are related to the fact that our graphsare unrooted. Trees and forests.
Designing a Boltzmann sampler for rooted trees is a basic exercise afterreading [18]. Note that the calculation of T ( x ) can be avoided by feeding the sampler directlywith the parameter t = T ( x ) , taken in (0 , . To sample unrooted trees, a first solution is tosample a rooted tree G and keep it with probability / | G | . However, this sometimes generateslarge rooted trees that are rejected with high probability. A much better solution is presentedin [13, Sec. 2.2.1]. In order to obtain an unrooted tree distributed according to (65), one calls SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 33 the sampler of rooted trees with a random parameter t . The density of t must be chosen to be (1 − t ) /C ( x ) on [0 , T ( x )] , with C ( x ) = T ( x ) − T ( x ) / . To sample t according to this density,we set t = 1 − (cid:112) − uC ( x ) , where u is uniform in [0 , . Again, we actually avoid computingthe series C ( x ) by feeding directly our sampler with the value T ( x ) ∈ (0 , . We use this trickfor all classes that involve the series T ( x ) .To obtain large forests (Figure 2), we actually sample forests with a distinguished vertex; thatis, a rooted tree plus a forest [18, Sec. 6.3]. Paths, cycles and stars.
The sequence operator of [18, Sec. 4] produces directed paths, whilewe need undirected paths. Let u be uniform on [0 , . Our generator generates the one-vertexpath if u < x/C p ( x ) , where C p ( x ) is given by (37), and otherwise generates a path of length x ) . An alternative is to generate a directed path, and reject it with probability / ifits size is at least 2.Although the cycle operator of [18, Sec. 4] generates oriented cycles, this does not create asimilar problem for our non-oriented cycles: indeed, a cycle of length at least 3 has exactly twopossible orientations.Designing a Boltzmann sampler Γ RS for rooted stars is elementary. For unrooted stars, wesimply call Γ RS , but reject the star with probability / if it has size 2 (because the only starwith two rootings has size 2). Graphs avoiding the bowtie.
This is the most complex of our algorithms, because thegeneration of connected graphs involves 7 different cases (see the proof of Proposition 30). Thereis otherwise no particular difficulty. We specialize this algorithm to the generation of graphsavoiding the 2-spoon (Proposition 11). However, the probability to obtain a forest is about . ,and thus there is no point in drawing a random graph of this class.The graphs shown in the paper have been drawn with the graphviz software.11.2. The nature of the dominant singularities of C ( z ) This is clearly a crucial point, as the probability that G n is connected and the quantities N n and S n seem to be directly correlated to it (see the summary of our results in Table 1).This raises the following question: is it possible to describe an explicit correlation between theproperties of the excluded minors and the nature of the dominant singularities of C ( z ) ? Forinstance, it is known that C ( ρ ) is finite when all excluded minors are 2-connected, but Section 4shows that this happens as well with some non-2-connected excluded minors. Which excludedminors give rise to a simple pole in C ( z ) (as in caterpillars)? or to a logarithmic singularity (asfor graphs with no bowtie nor diamond), or to a singularity in (1 − z/ρ ) − / (as for graphs withno bowtie)?Some classes for which C ( z ) has a unique dominant pole of high order are described in thenext subsection.11.3. More examples and predictions
Our examples, as well as a quick analysis, lead us to predict the following results when C ( z ) has a unique dominant singularity and a singular behaviour of the form (1 − z/ρ ) − α , with α > : • the mean and variance of the number N n of components scale like n α/ (1+ α ) , and N n admits a gaussian limit law after normalization, • the mean of S n scales like n / (1+ α ) , and S n , normalized by its expectation, converges toa Gamma distribution of parameters α + 1 and .The second point is developed in [10]. To confirm these predictions one could study the followingclasses, which yield series C ( z ) with a high order dominant pole. Fix k ≥ , and consider theclass A ( k ) of forests of degree at most k , in which each component has at most one vertex ofdegree ≥ . This means that the components are stars with long rays and “centers” of degree at most k . It is not hard to see that C ( k ) ( z ) = z + z − z ) + k (cid:88) i =3 z i +1 i !(1 − z ) i , so that C k has a pole of order k (for k ≥ ). The case k = 2 corresponds to forests of paths(Section 8). The limit case k = ∞ (forests of stars with long rays) looks interesting, with a veryfast divergence of C at 1: C ( ∞ ) ( z ) = z exp (cid:18) z − z (cid:19) − z − z ) . We do not dare any prediction here.11.4.
Other parameters
We have focussed in this paper on certain parameters that are well understood when allexcluded minors are 2-connected. But other parameters — number of edges, size of the largest 2-connected component, distribution of vertex degrees — have been investigated in other contexts,that sometimes intersect the study of minor-closed classes [8, 7, 14, 15, 20]. When specialized tothe theory of minor-closed classes, these papers generally assume that all excluded minors are2-connected, sometimes even 3-connected.Clearly, it would not be hard to keep track of the number of edges in our enumerative results.Presumably, keeping track of the number of vertices of degree d for any (fixed) d would not betoo difficult either. This may be the topic of future work. The size of the largest componentclearly needs a further investigation as well. Acknowledgements.
KW would like to thank Colin McDiarmid for many inspiring and helpfuldiscussions as well as for constant support. We also thank Philippe Duchon and Carine Pivoteaufor their help with Boltzmann samplers, Nicolas Bonichon for a crash course on the Graphvizsoftware, Jean-François Marckert for pointing out the relevance of [2] and finally Bruno Salvyfor pointing out Reference [25].We also thank the referees for their careful and detailed reports. (cid:47) (cid:1) (cid:5) (cid:3) (cid:46)
Appendix: Hayman-admissibility for bowties
Proposition 40.
Let C ( z ) and A ( z ) be the series given in Proposition 30. Then the series A ( z ) and C ( i +1) ( z ) A ( z ) are H-admissible, for any i ≥ .Proof. We begin with the series A ( z ) . Recall the analytic properties of T ( z ) , listed at thebeginning of the proof of Proposition 37. The capture condition H is readily checked. In fact, a ( r ) = rC (cid:48) ( r ) ∼ e − / √ − re ) / and b ( r ) = rC (cid:48) ( r ) + r C (cid:48)(cid:48) ( r ) ∼ √ e − / − re ) / . (66)Let us now prove H . By Taylor’s formula applied to the function f : θ (cid:55)→ C ( re iθ ) , we have,for r ∈ (0 , /e ) and θ ∈ [ − θ , θ ] : (cid:12)(cid:12) C ( re iθ ) − C ( r ) − iθa ( r ) + θ b ( r ) / (cid:12)(cid:12) ≤ | θ | / | α |≤ θ | f (3) ( α ) | with (cid:12)(cid:12)(cid:12) f (3) ( α ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) − ire iα C (cid:48) ( re iα ) − ir e iα C (cid:48)(cid:48) ( re iα ) − ir e iα C (cid:48)(cid:48)(cid:48) ( re iα ) (cid:12)(cid:12) ≤ rC (cid:48) ( r ) + 3 r C (cid:48)(cid:48) ( r ) + r C (cid:48)(cid:48)(cid:48) ( r ) ∼ κ (1 − re ) / SYMPTOTIC PROPERTIES OF SOME MINOR-CLOSED CLASSES OF GRAPHS 35 as r → /e , for some constant κ . Hence, if we take θ ≡ θ ( r ) = o ((1 − re ) / ) , then sup | θ |≤ θ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) A ( re iθ ) A ( r ) e − iθa ( r )+ θ b ( r ) / − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e o (1) − (cid:12)(cid:12)(cid:12) → as r → /e . Thus H holds for such values of θ . We now take θ ( r ) = (1 − re ) / , (67)and want to prove that H also holds.Recall that C ( z ) is analytic on C \ [1 /e, ∞ ) , and let us isolate in C ( z ) the part that divergesat z = 1 /e : C ( z ) = c √ − ze + 14 log 11 − ze + O (1) (68)where c = ( e − / / √ > . It follows that B ( z ) := C ( z ) − c √ − ze −
14 log 11 − ze is uniformly bounded on {| z | < /e } . Hence, writing z = re iθ , we have sup | θ |∈ [ θ ,π ) (cid:12)(cid:12)(cid:12)(cid:12) A ( z ) A ( r ) (cid:112) b ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M sup | θ |∈ [ θ ,π ) (cid:12)(cid:12)(cid:12)(cid:12) − re − ze (cid:12)(cid:12)(cid:12)(cid:12) / (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) c √ − ze − c √ − re (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:112) b ( r ) for some constant M .For any z of modulus r < /e , we have | − re | ≤ | − ze | , and we can bound the first factorabove by 1. Also, it is not hard to prove that (cid:60) (cid:0) / √ − ze (cid:1) is a decreasing function of θ ∈ (0 , π ) .Hence, denoting z = re iθ : sup | θ |∈ [ θ ,π ) (cid:12)(cid:12)(cid:12)(cid:12) A ( z ) A ( r ) (cid:112) b ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M exp (cid:18) (cid:60) (cid:18) c √ − z e (cid:19) − c √ − re (cid:19) (cid:112) b ( r ) . But as r → /e , the choice (67) of θ implies that (cid:60) (cid:18) c √ − z e (cid:19) − c √ − re = − c − re ) / + o (1) . Condition H now follows, using the estimate (66) of b ( r ) .Let us now consider the series A i ( z ) := C ( i +1) ( z ) A ( z ) , for i ≥ . It is easy to prove byinduction on i that for i ≥ , C ( i ) ( z ) = (2 i )!( e − / e i i √ i !(1 − ze ) i +1 / + O (cid:18) − ze ) i (cid:19) . (69)This can be proved either from the expression of C ( z ) given in Proposition 30, or by startingfrom the singular expansion (68) of C ( z ) and applying [19, Thm. VI.8, p. 419].Recall the behaviour (66) of the functions a ( r ) and b ( r ) associated with A ( z ) . It follows, withobvious notation, that as r → /e , a i ( r ) = a ( r ) + r C ( i +2) ( r ) C ( i +1) ( r ) = a ( r ) + O (cid:18) − re (cid:19) (70)and b i ( r ) = b ( r ) + r C ( i +2) ( r ) C ( i +1) ( r ) + r C ( i +3) ( r ) C ( i +1) ( r ) − r (cid:18) C ( i +2) ( r ) C ( i +1) ( r ) (cid:19) = b ( r ) + O (cid:18) − re ) (cid:19) (71)both tend to infinity. Thus H holds.Let us now prove that A i ( z ) satisfies H with the same value of θ as for A ( z ) (that is, θ = (1 − re ) / ). Thanks to (70–71) we have, for | θ | ≤ θ and uniformly in θ , e − iθa i ( r )+ θ b i ( r ) / = e − iθa ( r )+ θ b ( r ) / (cid:16) O ((1 − re ) / ) (cid:17) . Now using (69), we obtain, denoting z = re iθ , C ( i +1) ( z ) C ( i +1) ( r ) = (cid:18) − ze − re (cid:19) − i − / (cid:16) O ((1 − re ) / ) (cid:17) = 1 + O ((1 − re ) / ) . Hence A i ( z ) A i ( r ) e − iθa i ( r )+ θ b i ( r ) / = A ( z ) A ( r ) e − iθa ( r )+ θ b ( r ) / (cid:16) O ((1 − re ) / ) (cid:17) and Condition H holds for A i since it holds for A .Finally, since C ( z ) has non-negative coefficients, we have (cid:12)(cid:12) C ( i +1) ( z ) (cid:12)(cid:12) ≤ C ( i +1) ( r ) for z = re iθ .Thus the fact that A i ( z ) satisfies H follows from the fact that A ( z ) satisfies H , together with b i ( r ) ∼ b ( r ) . References [1] L. Addario Berry, C. McDiarmid, and B. Reed. Connectivity for bridge-addable monotone graph classes.
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MBM: CNRS, LaBRI, UMR 5800, Université de Bordeaux, 351 cours de la Libération, 33405Talence Cedex, France
E-mail address : [email protected] KW: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG,United Kingdom
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