Hans Jürgen Prömel

A New Approximation Algorithm for the Steiner Tree Problem with Performance Ratio 5/3

In this paper we present an RNC approximation algorithm for the Steiner tree problem in graphs with performance ratio 5/3 and RNC approximation algorithms for the Steiner tree problem in networks with performance ratio 5/3+? for all ?0. This is achieved by considering a related problem, the minimum spanning tree problem in weighted 3-uniform hypergraphs. For that problem we give a fully polynomial randomized approximation scheme. Our approach also gives rise to conceptually much easier and faster (though randomized) sequential approximation algorithms for the Steiner tree problem than the currently best known algorithms from Karpinski and Zelikovsky which almost match their approximation factor.

Excluding induced subgraphs: quadrilaterals

In this note we determine the structure of “almost all” graphs not containing a quadrilateral (i.e., a cycle of length four) as an induced subgraph. In particular, it turns out that there are asymptotically twice as many graphs not containing an induced quadrilateral than there are bipartite graphs.

On random planar graphs, the number of planar graphs and their triangulations

Let Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that |Pn| ≤ n! (75.8)n+o(n) and Bender, Gao and Wormald proved that |Pn| ≥ n! (26.1)n+o(n). Gerke and McDiarmid proved that almost all graphs in Pn have at least 13/7n edges. In this paper, we show that |Pn| ≤ n! (37.3)n+o(n) and that almost all graphs in Pn have at most 2.56n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least e3(3n-m)/2 labelled triangulations on n vertices, where e is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible.

RNC-Approximation Algorithms for the Steiner Problem

In this paper we present an RNC-algorithm for finding a minimum spanning tree in a weighted 3-uniform hypergraph, assuming the edge weights are given in unary, and a fully polynomial time randomized approximation scheme if the edge weights are given in binary. From this result we then derive RNC-approximation algorithms for the Steiner problem in networks with approximation ratio (1+e) 5/3 for all e>0.

Almost all Berge Graphs are Perfect

Let Per f(n) denote the set of all perfect graphs on n vertices and let Berge(n) denote the set of all Berge graphs on n vertices. The strong perfect graph conjecture states that Per f(n) = Berge(n) for all n . In this paper we prove that this conjecture is at least asymptotically true, i.e. we show that

Excluding induced subgraphs II: extremal graphs

Abstract In this paper we study properties of the classes of graphs not containing a fixed subgraph H as an induced subgraph. In particular, we introduce a new parameter τ(H) and show that fundamental results of extremal graph theory for weak subgraphs carry over to induced subgraphs, if one replaces in the corresponding theorems the chromatic number by τ(H).

THE ASYMPTOTIC NUMBER OF GRAPHS NOT CONTAINING A FIXED COLOR-CRITICAL SUBGRAPH

For a finite graphG letForb(H) denote the class of all finite graphs which do not containH as a (weak) subgraph. In this paper we characterize the class of those graphsH which have the property that almost all graphs inForb(H) are ℓ-colorable. We show that this class corresponds exactly to the class of graphs whose extremal graph is the Turán-graphTn(ℓ).An earlier result of Simonovits (Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions,Discrete Math.7 (1974), 349–376) shows that these are exactly the (ℓ+1)-chromatic graphs which contain a color-critical edge.

The average number of linear extensions of a partial order

Kleitman and Rothschild ( Trans. Amer. Math. Soc. 205 (1975), 205–220) gave an asymptotic formula for the number of partial orders with ground-set [ n ]. We give a shorter proof of their result and extend it to count the number of pairs ( P , ≺), where P is a partial order on [ n ] and ≺ is a linear extension of P . This gives us an asymptotic formula for (a) the average number of linear extensions of an n -element partial order and (b) the number of suborders of an n -element linear order.

Induced Ramsey Numbers

We investigate the induced Ramsey number of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with , we have where is the chromatic number of H and C is some universal constant. Furthermore, we also investigate imposing some conditions on G. For instance, we prove a bound that is polynomial in both k and t in the case in which G is a tree. Our methods of proof employ certain random graphs based on projective planes.

Counting unlabeled structures

Abstract In this note we prove that whenever l is an infinite class of finite labeled structures provided with one binary relation such that l is closed under isomorphisms and (induced) substructures and l is rich enough (in a quantitative sense) then almost all structures in l are rigid, i.e., have no nontrivial automorphism. Applying this result to well-known results for labeled graphs we derive, for example, that almost every unlabeled Kl+1-free graph is already l-colorable, and we obtain 0–1 laws for the classes of unlabeled Kl+1-free graphs. It is worth while to note that a special case of our result states that almost all partial orders are rigid. As a consequence of this and the Kleitman-Rothschiid theorem (Trans. Amer. Math. Soc. 205 (1975), 205–220) we get an asymptotic formula for the number of unlabeled partial orders.