On the probability of planarity of a
random graph near the critical point
Let be the uniform random graph with vertices and edges. Erdős and Rényi (1960) conjectured that the limiting probability
exists and is a constant strictly between and . Łuczak, Pittel and Wierman (1994) proved this conjecture and Janson, Łuczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point . For each , we find an exact analytic expression for
In particular, we obtain . We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of being series-parallel converges to . For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.
We dedicate this paper to the memory of Philippe Flajolet.
The random graph model assigns uniform probability to graphs on labelled vertices with edges. A fundamental result of Erdős and Rényi  is that the random graph undergoes an abrupt change when is around , the value for which the average vertex degree is equal to one. When and , almost surely the connected components are all of order , and are either trees or unicyclic graphs. When and , almost surely there is a unique giant component of size . We direct to reader to the reference texts  and  for a detailed discussion of these facts.
We concentrate on the so-called critical window where is a real number, identified by the work of Bollobás [2, 3]. Let us recall that the excess of a connected graph is the number of edges minus the number of vertices. A connected graph is complex if it has positive excess. As , complex components disappear and only trees and unicyclic components survive, and as , components with unbounded excess appear. A thorough analysis of the random graph in the critical window can be found in  and , which constitute our basic references.
For each fixed , we denote the random graph by . The core of is obtained by repeatedly removing all vertices of degree one from . The kernel is obtained from by replacing all maximal paths of vertices of degree two by single edges. The graph satisfies almost surely several fundamental properties, that were established in  by a subtle simultaneous analysis of the and the models.
The number of complex components is bounded.
Each complex component has size of order , and the largest suspended tree in each complex component has size of order .
has size of order and maximum degree three, and the distance between two vertices of degree three in is of order .
is a cubic (3-regular) multigraph of bounded size.
The key property for us is the last one. It implies that almost surely the components of are trees, unicyclic graphs, and those obtained from a cubic multigraph by attaching rooted trees to the vertices of , and attaching ordered sequences of rooted trees to the edges of . Some care is needed here, since the resulting graph may not be simple, but asymptotically this can be accounted for.
It is clear that is planar if and only if the kernel is planar. Then by counting planar cubic multigraphs it is possible to estimate the probability that is planar. To this end we use generating functions. The trees attached to are encoded by the generating function of rooted trees, and complex analytic methods are used to estimated the coefficients of the corresponding series. This allows us to determine the exact probability
In particular, we obtain .
This approach was initiated in the seminal paper by Flajolet, Knuth and Pittel , where the authors determined the threshold for the appearance of the first cycles in . A basic feature in  is to estimate coefficients of large powers of generating functions using Cauchy integrals and the saddle point method. This path was followed by Janson, Knuth, Łuczak and Pittel , obtaining a wealth of results on . Of particular importance for us is the determination in  of the limiting probability that has given excess. The approach by Łuczak, Pittel and Wierman in  is more probabilistic and has as starting point the classical estimates by Wright  on the number of connected graphs with fixed excess. The range of these estimates was extended by Bollobás  and more recently the analysis was refined by Flajolet, Salvy and Schaeffer , by giving complete asymptotic expansions in terms of the Airy function.
The paper is organized as follows. In Section 2 we present the basic lemmas needed in the sequel. Except for the proof of Lemma 1, the paper is self-contained. Lemmas 2 to 5 are proved in  with a different presentation; for the sake of completeness and exposition we provide shorter and hopefully more accessible proofs. In Section 3 we compute the number of cubic planar multigraphs, suitably weighted, where we follow . In Section 4 we compute the exact probability that the random graph is planar as a function of . We generalize this result by determining the probability that belongs to a minor-closed class of graphs in several cases of interest.
We close this introduction with a remark. The problem of 2-satisfiability presents a striking analogy with the random graph process. Given Boolean variables and a conjunctive formula of clauses, each involving two literals, the problem is to determine the probability that the formula is satisfiable when grows with . The threshold has been established at and the critical window is also of width ; see . However the exact probability of satisfiability when the number of clauses is has not been determined, and appears to be a more difficult problem.
All graphs in this paper are labelled. The size of a graph is its number of vertices. A multigraph is a graph with loops and multiple edges allowed.
We recall that the generating function of rooted trees satisfies
Using Lagrange’s inversion , one recovers the classical formula for the number of rooted labelled trees.The generating function for unrooted trees is
This can be proved by integrating the relation , or more combinatorially using the dissimilarity theorem for trees .
A graph is unicyclic if it is connected and has a unique cycle. Unicyclic graphs can be seen as an undirected cycle of length at least three to which we attach a sequence of rooted trees. Since the directed cycle construction corresponds algebraically to (see ), the generating function is
Graphs all whose components are unicyclic are given by the exponential formula:
The following result, which is fundamental for us, is proved in [16, Theorem 4] by a careful analysis of the structure of complex components in . We say that a property holds asymptotically almost surely (a.a.s.) in if the probability that is satisfied tends to one as . Recall that .
For each , the kernel of is a.a.s. a cubic multigraph.
Given a cubic multigraph with loops, double edges and triple edges, define its weight as
This weight (called the compensation factor in ), has the following explanation. When we substitute edges of the kernel by sequences of rooted trees, a loop has two possible orientations that give raise to the same graph. A double (triple) edge can be permuted in two (six) ways, again producing the same graph. From now on, all multigraphs we consider are weighted, so that we omit the qualifier. The following lemma is proved in  using a combination of guessing and recurrence relations. The next proof is already contained in [4, Chap. 2].
The number of cubic multigraphs with vertices is equal to
A cubic multigraph can be modeled as a pairing of darts (half-edges), 3 for each vertex, with a total of darts. The number of such pairings is . However, we have to divide by the number of ways of permuting each of the triples of darts. The weight takes care exactly of the number of times a cubic multigraph is produced in this process. ∎
The next result is essentially proved in  using several algebraic manipulations. Here we present a concise proof. We denote by the coefficient of in the power series .
The number of simple graphs with vertices, edges and cubic kernel of size satisfies
Such a graph is the union of a set of unrooted trees, a set of unicyclic graphs, and a cubic multigraph with a rooted tree attached to each vertex of and a sequence (possibly empty) of rooted trees attached to each edge of . Let us see first that . Indeed, the final excess of edges over vertices must be . Each tree component contributes with excess , each unicyclic component with excess , and (together with the attached trees) with excess . Hence
The first two factors and on the right-hand side of the inequalities encode the set of trees and unicyclic components. The last part encodes the kernel . It has vertices and is labelled, hence the factor ; the weighting guarantees that each graph contributing towards is counted exactly once. The trees attached to the vertices give a factor . The sequences of trees attached to the edges give each a factor . However, this allows for the empty sequence and the resulting graph may not be simple, so we get only an upper bound. To guarantee that the final graph is simple we take sequences of length at least two, encoded by (length one is enough for multiple edges of , but length two is needed for loops). Since this misses some graphs, we get a lower bound. ∎
Let . Then for any fixed and integer we have
uniformly for , where
where is the polynomial
and is a path in the complex plane that consists of the following three straight line segments:
The constant term in the left-hand side of (1) is estimated using Stirling’s formula, getting
The coefficient of in Equation (1) is estimated by means of a contour integral around , using the expressions of and in terms of :
We make the change of variable , whose inverse is , and we obtain
where the integrand is split into a smooth function
and a large power involving
The contour path in Equation (4) should be such that . As remarked in  (see also ), the function satisfies . Moreover, precisely at the critical value we also have . This triple zero shows up in the procedure used in  when estimating (4) for large by means of the saddle-point method. Notice that is singular at , due to the singularity at of .
Let , and let be the positive solution to
This choice is necessary in order to get precise bounds for the tail estimates that appear using the saddle-point method. Following the proof of [13, Lemma 3], we evaluate (4) on the path , where runs from to . That is,
The main contribution to the value of this integral comes from the vicinity of . The magnitude of depends on the real part . Observe that decreases as increases, and that has its maximum on the circle when . We write . Analyzing , we have
uniformly in any region such that . For the function , we have
If is the integrand of (4), we have
when . Finally,
It is important to notice that in the previous lemma the final asymptotic estimate does not depend on the choice of . The next result is a direct consequence and can be found as Formula (13.17) in .
The limiting probability that the random graph has a cubic kernel of size is equal to
where and is as in the previous lemma.
In particular, for the limiting probability is
Using the notation of Lemma 3, the probability for a given is by definition
thus proving the result. A key point is that the discrepancy between the factors and in the bounds for does not affect the limiting value of the probability. ∎
Notice that if we replace the by the numbers arising by counting planar cubic multigraphs, we obtain immediately the probability that has a cubic planar kernel of size . Since is planar if and only if its kernel is planar, we can use this fact to compute the probability of being planar. But first we must compute .
3 Planar cubic multigraphs
In this section we compute the numbers of cubic planar multigraphs of size . The associated generating function has been obtained recently by Kang and Łuczak  (generalizing the enumeration of simple cubic graphs in ), but their derivation contains some minor errors. They do not affect the correctness of , since the asymptotic estimates needed by the authors are still valid. However, for the computations that follow we need the exact values. The next result is from , with the corrections mentioned below. All multigraphs are weighted as in the previous section.
Let be the generating function of connected cubic planar multigraphs. Then is determined by the following system of equations:
The generating functions and correspond to distinct families of edge-rooted cubic planar graphs, and is an algebraic function related to the enumeration of 3-connected cubic planar graphs (dually, 3-connected triangulations).
The corrections with respect to  are the following. In the first equation a term has been removed. In the second and sixth equations we have replaced a term by . In the fourth equation we have removed a term . For the combinatorial interpretation of the various generating functions and the proof of the former equations we refer to . Notice that eliminating from the last two equations we obtain a relation between and . This relation can be used to obtain a single equation satisfied by , eliminating from the remaining equations. We reproduce it here in case the reader wishes to check our computations.
The first terms are
This allows us to compute and , hence also . The first coefficients of are as follows.
Using the set construction, the generating function for cubic planar multigraphs is then
where is the number of planar cubic multigraphs with vertices. This coincides with the generating function for all cubic (non-necessarily planar) multigraphs up to the coefficient of . The first discrepancy is in the coefficient of . The difference between the coefficients is , corresponding to the 10 possible ways of labelling , the unique non-planar cubic multigraph on six vertices.
4 Probability of planarity and generalizations
Let be a graph with a cubic kernel . Then clearly is planar if and only if is planar, and we can compute the probability that is planar by counting over all possible planar kernels.
Let be the number of cubic planar multigraphs with vertices. Then the limiting probability that the random graph is planar is
In particular, the limiting probability that is planar is
The same analysis as in Section 2 shows that is the probability that the kernel is planar and has vertices. Summing over all possible , we get the desired result. ∎
obtained by considering connected cubic multigraphs with at most six vertices. We remark that Stepanov  showed that for (without actually establishing the existence of the limiting probability). The function is plotted in Figure 1. As expected, is close to 1 when and close to 0 when . For instance, and .
Besides planar graphs, one can consider other classes of graphs. Let be a class of graphs closed under taking minors, that is, if is a minor of and , then . If are the excluded minors of , then we write . (By the celebrated theorem of Robertson and Seymour, the number of excluded minors is finite, but we do not need this deep result here). The following result generalizes the previous theorem.
Let and assume all the are 3-connected. Let be the number of cubic multigraphs in with vertices. Then the limiting probability that the random graph is in is
In particular, the limiting probability that is in is
Moreover, for each we have
If all the are 3-connected, then clearly a graph is in if and only its kernel is in . The probability is then computed as in Theorem 7. It is positive since contains all trees and unicyclic graphs, which contribute with positive probability (although tending to 0 as ). To prove that it is less than one, let be the largest size of the excluded minors . By splitting vertices it is easy to construct cubic graphs containing as a minor, hence contains as a minor with positive probability (alternatively, see the argument at the end of ). It follows that . ∎
In some cases of interest we are able to compute the numbers explicitly. Let be the class of series-parallel graphs. The same system of equations as in Lemma 6 holds for series-parallel graphs with the difference that now (this is due to the fact that there are no 3-connected series-parallel graphs). The generating function for cubic series-parallel multigraphs can be computed as
For instance, , corresponding to the fact that is the only cubic multigraph with vertices which is not series-parallel. The limiting probability that is series-parallel is
See Figure 1 for a plot of .
The class of outerplanar graphs does not fall directly under this scheme, since is not 3-connected, but adapting the equations in Lemma 6 (in particular the parallel decomposition encoded by ) it is possible to enumerate exactly cubic outerplanar multigraphs. The first terms in the generating function are
The first discrepancy with is at , corresponding to the graph with either a loop or a double edge attached at the vertices of degree two. The probability of being outerplanar is
We do not plot in Figure 1 since it is too close to to see a clear distinction.
As another example, consider excluding . Since the only 3-connected non-planar graph in is , which is not cubic, the limiting probability of being in this class is exactly the same as of being planar, although is exponentially larger than the class of planar graphs . But excluding the graph , obtained by adding one edge to , does increase the probability, since is in the class and is cubic and non-planar (the probability is computable since the 3-connected graphs in are known ). Other classes such as or can be analyzed too using the results from .
It would be interesting to compute the probability that has genus . For this we need to count cubic multigraphs of genus (orientable or not). We only know how to do this for , the reason being that a 3-connected planar graph has a unique embedding in the sphere. This is not at all true in positive genus. It is true though that almost all 3-connected graphs of genus have a unique embedding in the surface of genus (see ). This could be the starting point for the enumeration, by counting first 3-connected maps of genus (a map is a graph equipped with a 2-cell embedding). But this is not enough here, since we need the exact numbers of graphs.
The first and third authors acknowledge the warm hospitality and support from the Laboratoire d’Informatique Algorithmique: Fondements et Applications (LIAFA) in Paris, where much of this work was done.
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