5.3 Random outerplanar graphs
5.3.3 Chromatic number
It is easy to see that every outerplanar graphs is three colourable. Indeed more is true.
Theorem 5.3.4. Let χ(Gn) denote the chromatic number of a random outer-planar graph Gn onnvertices. Then we have
n→∞lim Pr(χ(Gn) = 3) = 1.
This follows from the fact that the number of labeled bipartite outerplanar graphs onnvertices is asymptoticallyc 4.40364nn! for a constantc >0, which was proven by Löffler [95].
5.4 Recursive counting and uniform sampling
Observe that the block structure of an outerplanar graph is a forest. Thus we can count and generate outerplanar graphs similarly as demonstrated for forests in Section 4.4. The decomposition from a (not-necessarily connected) outerplanar graphs to connected outerplanar graphs is followed by formulas (2.2.1) and (2.2.2). Thus we restrict our attention to connected outerplanar graphs.
In order to decompose a labeled connected outerplanar graph, we consider two cases. The vertex labeled with the smallest label is either a cutvertex and hence it is contained in more than one block, or it is not a cutvertex and hence it is contained in a unique block. Depending on these two cases, we apply a degree-reduction strategy as in the case of labeled trees (see Figure 5.1).
Let c(n) be the number of all labeled connected outerplanar graphs withn vertices {1,· · · , n} andcd(n) the number of all labeled connected outerplanar graphs with n vertices, where the vertex 1 is adjacent to d blocks. Then, for n≥2,
c(n) =
n−1
X
d=1
cd(n). Note that Pn−1
d=2cd(n) counts all labeled connected outerplanar graphs with n vertices where the vertex 1 is a cutvertex, and c1(n) all labeled connected outerplanar graphs withnvertices where the vertex 1 is not a cutvertex.
Let Gbe a labeled connected outerplanar graph with nvertices where the vertex 1 is a cutvertex and is adjacent to dblocks, d≥2; see the upper part
5.4. RECURSIVE COUNTING AND UNIFORM SAMPLING 61
Figure 5.1: Pulling off the petals from the flower.
of Figure 5.1. At the vertex 1 we split off the connected component containing the vertex 2 fromG. In the remaining graph the vertex 1 is adjacent tod−1 blocks. If the split subgraph hasivertices, then there are n−2i−2
ways to choose a vertex set of the split subgraph since the two vertices 1 and 2 are already contained in the split subgraph. It follows that ford≥2,n≥3,
cd(n) =
n−d+1
X
i=2
n−2 i−2
c1(i)cd−1(n−i+ 1).
We consider the case that the vertex 1 is not a cutvertex and hence it is contained in a unique block, which we call the root block; see the lower part of Figure 5.1. Let qc(n) be the number of all labeled connected outerplanar graphs withn vertices, where the smallestc vertices of the root block are not cutvertices. Then clearlyc1(n) = q1(n).
From such a graph we split off a subgraph attached at the (c+1)-th smallest vertex, which might be any kind of outerplanar graph. Then in the remaining graph the (c+ 1)-th smallest vertex of the root block is not a cutvertex. Thus forc≥1,n≥3,
qc(n) =
n−b+1
X
i=1
n−1 i−1
c(i)qc+1(n−i+ 1).
If none of the vertices in the root block of an outerplanar graph with n vertices are not cutvertices, the graph is two-connected and thusqn(n) =bn.
We have a complete set of recursive formulas that count outerplanar graphs.
Table 5.1 shows the exact numbers bn, cn, and gn of labeled two-connected outerplanar graphs, connected outerplanar graphs, and outerplanar graphs, on nvertices up ton= 16.
62 CHAPTER 5. LABELED OUTERPLANAR GRAPHS
n bn cn gn
1 0 1 1
2 1 1 2
3 1 4 8
4 9 37 63
5 132 602 893
6 2700 14436 19714
7 70920 458062 597510
8 2275560 18029992 22903403
9 86264640 845360028 1056115331
10 3772681920 4593606320 56744710974
11 186972105600 2836966508216 3475626211316
12 10355595465600 196156795008384 238818544070905
13 633892275878400 15008752290350656 18183183610029003 14 42495895579737600 1258841795197091392 1519020289266947462 15 3096545573029708800 114838947237881287800 138117136134012654182 16 243680880958010496000 11319937495659268412416 13576724206357958780409
Table 5.1: The exact numbersbn,cn,gn of labeled two-connected outerplanar graphs, connected outerplanar graphs, outerplanar graphs onn vertices, up to n= 16.
Uniform sampling. The decomposition and counting formulas presented above give rise to an efficient uniform random generation procedure.
Our sampling procedure first determines the number of components, and how many vertices they shall contain. Each connected component is gener-ated independently from the others, but having the chosen numbers of vertices.
To generate a connected component with given numbers of vertices, we decide for a decomposition into 2-connected subgraphs and how the vertices shall be distributed among its parts. For the generation of two-connected outerplanar graphs we use the tree structure of its dual.
Theorem 5.4.1. Labeled outerplanar graphs onnvertices can be sampled uni-formly at random in deterministic time O˜(n4) and space O(n3logn). If we apply a preprocessing step, this can also be done in deterministic timeO˜(n2).
Brute-force algorithms to generate random outerplanar graphs uniformly at random require exponential time, and Markov chain Monte Carlo methods have unknown mixing times and only approximate the uniform distribution. We have developed a polynomial time generation algorithm for outerplanar graphs, which can be adapted to generate and count labeled outerplanar graphs, con-nected outerplanar graphs and two-concon-nected outerplanar graphs, uniformly at random. In all these cases, it is also easy to modify the counting formulas and the uniform sampling algorithm for outerplanar graphs with a given number of vertices and a given number of edges and also for outerplanar multigraphs.
The recursive counting formulas and the uniform sampling algorithm are imple-mented by Löffler [95].
Chapter 6
Labeled Cubic Planar Graphs
In this chapter we decompose labeled cubic planar graphs along the connectivity structure, and derive the asymptotic number by interpreting the decomposition in terms of generating functions and then by applying the singularity analysis.
For the decomposition, we make use of a rooted cubic graph with one dis-tinguished oriented edge, and decompose rooted connected cubic graphs into smaller parts up to rooted 3-connected cubic graphs. To complete the counting and generation procedure, it suffices to consider 3-connected cubic graphs, be-cause no cubic graph is 4-connected. For 3-connected cubic graphs, we can use their dual, i.e., triangulations.
Based on the decomposition, we derive the equations of generating functions and apply theresultant method suggested by Flajolet and Sedgewick [63]. We show that the number of labeled cubic planar graphs onnvertices is asymptot-icallyc n−7/2 ρ−n n!, for a suitable positive constantcandρ−1= 3. .132595.
Using the asymptotic number, we also study the typical properties of a random cubic planar graph that holds when the number of vertices converges to infinity, e.g., the chromatic number. To this end, we first show that the number of isolated K4’s in a random cubic planar graph has asymptotically Poisson distribution with mean ρ4/4! and that a random cubic planar graph contains linearly many triangles with probability tending to one. As a consequence, together with Brooks’ theorem, we can see that the chromatic number of a random cubic planar graph is four with probability bounded away from zero and one, and that the chromatic number of every connected component with more than four vertices in a random cubic planar graph is three with probability tending to one.
Using a complete set of recursive counting formulas, we derive a deterministic uniform generation of cubic planar graphs from the general principle. Further-more, we can compute the exact numbers of cubic planar graphs according to the connectivity computed from the recursive enumeration.
The rest of the chapter is organized as follows: In Section 6.1, we introduce necessary terminologies and the decomposition theorem for rooted cubic planar graphs. In Section 6.2, we interpret the decomposition in terms of generating functions. In Section 6.3, we provide the relation between 3-connected cubic
pla-63
64 CHAPTER 6. LABELED CUBIC PLANAR GRAPHS
nar graphs and triangulations and derive counting formulas for triangulations.
In Sections 6.4 and 6.5, we derive the equations of generating functions, get the asymptotic number of labeled cubic planar graphs, and study some properties of a random cubic planar graph. In Section 6.6, we derive recursive counting for-mulas based on the decomposition theorem, and discuss the uniform generation algorithm.