Three-connected cubic planar graphs

also forn= 0,2 except that we setg⁽⁰⁾₀ = 1 by convention.

If we select an arbitrary edge in a connected cubic planar (simple) graph and orient this edge, we obtain a rooted cubic graphG= (V, E, st) that is neither a b-graph, nor ans- orp-graph wheresandtare adjacent in the underlying graph G⁻, see Figure 6.7. Note that the number of connected cubic planar (simple) graphs with one distinguished oriented edge is counted by 3x^dG(1)_dx^(x), and the number ofs- (resp. p-)graphsG= (V, E, st) wheresandt are adjacent inG⁻ as depicted in the middle (resp. right) picture in Figure 6.7 is counted byB(x)² (resp. x²C(x)). Therefore we get

3xdG⁽¹⁾(x)

dx = D(x) +S(x) +P(x) +H(x)−B(x)²−x²C(x). (6.2.2)

Figure 6.7: Types of rooted cubic graphs that are not simple.

As we have seen in (2.2.3), the exponential generating function for connected cubic planar graphs and that for not necessarily connected ones are related as follows.

G⁽⁰⁾(x) = exp(G⁽¹⁾(x)). (6.2.3)

6.3 Three-connected cubic planar graphs

The number of labeled three-connected cubic planar graphs is closely related to that of rooted triangulations. A rooted triangulation is an edge-maximal plane graph with a distinguished directed edge on the outer face, called the root edge. Tutte [137] derived exact and asymptotic formulas for the number of such objects up to isomorphisms that preserve the outer face and the root edge. Since such graphs do not have non-trivial automorphisms that fix the root edge, we can obtain the number of labeled objects from the number of unlabeled objects. Labeled three-connected planar graphs with at least four vertices have exactly two non-equivalent embeddings in the plane. Using plane duality, we can compute the number of rooted three-connected cubic planar graphs from the number of rooted triangulations.

Let tn be the number of unlabeled rooted triangulations onn+ 2 vertices.

From the formulas Tutte computed for unlabeled rooted triangulations onn+ 3 vertices, it follows that the ordinary generating functionT(z) fortn, i.e.,T(z) =

70 CHAPTER 6. LABELED CUBIC PLANAR GRAPHS

P

n≥1t_nzⁿ, satisfies the following.

T(z) =u(1−2u) (6.3.1)

z=u(1−u)³.

The first terms ofT(z) arez+z²+ 3z³+ 13z⁴+ 68z⁵+ 399z⁶+. . .. Further, T(z) has a dominant singularity atξ= 27/256 and the asymptotic growth oftn

isα4n^−5/2ξ⁻ⁿn!, whereα4is a constant. Let ˜T(x, y) be the corresponding ordi-nary generating function, but wherexmarks two times the number of faces and ymarks three times the number of edges. By Euler’s formula, a triangulation on n+ 2 vertices has 2nfaces and 3nedges. Therefore, ˜T(x, y) :=P

n≥1t_nx²ⁿy³ⁿ can be computed by ˜T(x, y) =T(x²y³).

We now determine the exponential generating functionM(x, y) for the num-ber of labeled rooted 3-connected cubic graphs, which was needed in the decom-position of h-graphs in Section 6.1. Since the dual of a 3-connected cubic map on 2nvertices is a triangulation onn+ 2 vertices (and hence with 2nfaces and 3nedges), we have m_2n,3n= (2n)!t_n/2 forn≥2. We therefore obtain

M(x, y) =X

n≥2

m2n,3n

(2n)! x²ⁿy³ⁿ =1

2( ˜T(x, y)−x²y³) = 1

2(T(x²y³)−x²y³). (6.3.2) Thus M(x, y) = (x⁴y⁶ + 3x⁶y⁹+ 13x⁸y¹² + 68x¹⁰y¹⁵ + 399x¹²y¹⁸+. . .)/2.

Furthermore the dominant singularity ofM(x) =M(x,1) = 1/2 (T(x²)−x²) is the square-root of the dominant singularity ofT(z) and the asymptotic growth ofm_n withneven isα₃n^−5/2θ⁻ⁿn!, whereθ= 3√

3/16 andα₃ is a constant.

6.4 Singularity analysis

We summarize the equations derived so far.

B(x) =x²(D(x) +C(x)−B(x))/2 (6.4.1) C(x) =S(x) +P(x) +H(x) +B(x) (6.4.2)

D(x) =B(x)²/x² (6.4.3)

S(x) =C(x)²−C(x)S(x) (6.4.4) P(x) =x²C(x) +x²C(x)²/2. (6.4.5) We can also describe the substitution in Equation (6.2.1) forH(x) algebraically, using Equations (6.3.1) and (6.3.2).

2(C(x) + 1)H(x) =u(1−2u)−u(1−u)³ (6.4.6) x²(C(x) + 1)³=u(1−u)³. (6.4.7) Using algorithms for computing resultants and factorizations (these are stan-dard procedures in e.g., Maple or Mathematica), we can obtain a single alge-braic equationQ(C(x), x) = 0 from equations (6.4.1) – (6.4.7) that describes the generating function C(x) uniquely, given sufficiently many initial terms of cn.

6.4. SINGULARITY ANALYSIS 71 This is in principle also possible for all other generating functions involved in the above equations; however, the computations turn out to be more tedious, whereas the computations to compute the algebraic equation forC(x) are man-ageable.

From this equation, following the discussion in Section VII.4 in [63], one can obtain the two dominant singularities ρ and −ρ of C(x), where ρ is an analytic constant and the first digits are ρ .

= 0.319224. We can also compute the expansion at the dominant singularityρ. Changing the variablesY =C(x)− C(ρ) and X = x−ρin Q(C(x), x) = 0, one can symbolically verify that the equationQ(C(x), x) = 0 can be written in the form

(aY +bX)²=pY³+qXY²+rX²Y +sX³+ higher order terms, where a, b, p, q, r, s are constants that are given analytically. This implies the following expansion ofC(x) near the dominant singularityρ.

C(x) =C(ρ) +bρ/a(1−x/ρ) +β1(1−x/ρ)^3/2+O((1−x/ρ)²), (6.4.8) whereβ₁:=ρ^3/2/ap

p(b/a)³−q(b/a)²+r(b/a)−s is a positive constant. For large n, the coefficientc⁺_n ofxⁿ on the right hand side satisfies

c⁺_n ∼ β2 n^−5/2 ρ⁻ⁿ n!, whereβ2=β1/Γ(3/2) = 2β1/√

π. Similarly we get the expansion at the domi-nant singularity−ρ

C(x) =C(ρ) +bρ/a(1 +x/ρ) +β₁(1 +x/ρ)^3/2+O((1 +x/ρ)²), and for large n, the coefficientc⁻_n ofxⁿ on the right hand side satisfies

c⁻_n ∼ β₂n^−5/2 (−ρ)⁻ⁿ n!.

Following Theorem VI.8 [63], the asymptotic numberc_nis then the summation of these two contributionsc⁺_n andc⁻_n, and thus for large evenn

cn ∼ 2β2 n^−5/2 ρ⁻ⁿ n!, whereasc_n= 0 for oddn.

Since the generating functions for B(x), D(x), S(x), P(x), H(x) are related with C(x) by algebraic equations, they all have the same dominant singulari-tiesρand−ρ. The singular expansion ofG⁽¹⁾(x) can be obtained from Equa-tion (6.2.2) through a term-by-term integraEqua-tion, and thus we obtain the singular expansions atρand−ρ

G⁽¹⁾(x) =G⁽¹⁾(ρ) +c(1−x/ρ)²+β3(1−x/ρ)^5/2+O((1−x/ρ)³), (6.4.9) G⁽¹⁾(x) =G⁽¹⁾(ρ) +c(1 +x/ρ)²+β₃(1 +x/ρ)^5/2+O((1 +x/ρ)³), (6.4.10) wherecandβ3are analytically given constants. Thus for an analytically given constantα1and for large even n we get

g_n⁽¹⁾ ∼ α₁n^−7/2ρ⁻ⁿ n!, whereasg⁽¹⁾n = 0 for oddn.

72 CHAPTER 6. LABELED CUBIC PLANAR GRAPHS

Because of Equation (6.2.3), the generating functions G⁽⁰⁾(x) and G⁽¹⁾(x) have the same dominant singularities ρand −ρ, and indeed we may see that gn⁽¹⁾/gn⁽⁰⁾→e^−λwhereλ=G⁽¹⁾(ρ). Based on the above decomposition it is also easy to derive equations for the exponential generating functionG⁽²⁾(x) for the number of biconnected cubic planar graphs, which has a slightly larger radius of convergenceη (whose first digits are 0.319521).

We finally obtain the following.

Theorem 6.4.1. The asymptotic number of labeled cubic planar graphs, labeled connected cubic planar graphs, labeled 2-connected cubic planar graphs, and la-beled 3-connected cubic planar graphs is given by the following. For large even n

g_n⁽⁰⁾ ∼α0 n^−7/2 ρ⁻ⁿ n! g_n⁽¹⁾ ∼α1 n^−7/2 ρ⁻ⁿ n! g_n⁽²⁾ ∼α₂ n^−7/2 η⁻ⁿ n! g_n⁽³⁾ ∼α3 n^−7/2 θ⁻ⁿ n!.

All constants are analytically given. Also α₁/α₀ = e^−λ where λ = G⁽¹⁾(ρ).

Furtherρ⁻¹ .

= 3.132595 ,η⁻¹ .

= 3.129684, andθ⁻¹ .

= 3.079201.