This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class -- an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.

A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six conjectures stating that if H has n vertices and does not contain p then the size of H is O(n) and the number of such Hs is O(c n). The latter part generalizes the Stanley--Wilf conjecture on permutations. Using generalized Davenport--Schinzel sequences, we prove the conjectures with weaker bounds O(nfi(n)) and O(fi(n) n), where fi(n) ! 1 very slowly. We prove the conjectures fully if p first increases and then decreases or if p

Abstract. For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “non-crossing marked graphs ” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni + n − 3 where ni is the number of points of A in the interior of conv(A). The vertices of this polytope are all the pseudo-triangulations of A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs. 1.

We investigate the number of plane geometric, i.e., straight-line, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ ∗ ( √ 72 n) = Θ ∗ (8.4853 n) triangulations and Θ ∗ (41.1889 n) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.

We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1-entries in an n × n (0,1)-matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Brändén and Mansour. We then extend the original Füredi–Hajnal problem from ordinary matrices to d-dimensional matrices and show that the number of 1-entries in a d-dimensional (0,1)-matrix with side length n which avoids a d-dimensional permutation matrix is O(n d−1).

Here we consider two parameters for random non-crossing trees:   i ¡ the number of random cuts to destroy a sizen non-crossing tree and   ii ¡ the spanning subtree-size of p randomly chosen nodes in a size-n non-crossing tree. For both quantities, we are able to characterise for n ¢ ∞ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter   ii ¡ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter   i ¡ , we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks. Keywords: Non-crossing trees, generating function, limiting distributions 1

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2 ρ −n, where g ≈ 0.00909941 and ρ −1 ≈ 7.50360 can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n vertices, for instance concerning connectedness, the chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.

We show that the number gn of labelled series-parallel graphs on n vertices is asymptotically gn ∼ g · n −5/2 γ n n!, where γ and g are explicit computable constants. We show that the number of edges in random series-parallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs.

In the past decades the Gn,p model of random graphs, introduced by Erdős and Rényi in the 60’s, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn,p appear independently. The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of Gn,p and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Rn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. Consequently, so far basically all results about properties of random graphs with structural side constraints rely on completely different methods, mostly from analytic combinatorics. In this paper we show that recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer (Combinatorics, Probability and Computing 13, 2004) and Fusy (International Conference on Analysis of Algorithms ’05) can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables – to which we can then again apply Chernoff bounds to obtain extremely tight results. We elaborate our ideas by studying random dissections and triangulations of a labeled convex n-gon. For both we obtain the degree sequence and the number of induced copies of given fixed graphs. The degree sequence for triangulations was already obtained previously by Gao and Wormald (Combinatorica 23, 2003) using deep methods from analytic combinatorics. We do, however, get better probabilities for the tails of the distributions. 1