P. Erdős

Compactness results in extremal graph theory

Let L be a given family of so called prohibited graphs. Let ex (n, L) denote the maximum number of edges a simple graph of ordern can have without containing subgraphs from L. A typical extremal graph problem is to determine ex (n, L), or at least, find good bounds on it. Results asserting that for a given L there exists a much smaller L*⫅L for which ex (n, L) ≈ ex (n, L*) will be calledcompactness results. The main purpose of this paper is to prove some compactness results for the case when L consists of cycles. One of our main tools will be finding lower bounds on the number of pathsPk+1 in a graph ofn vertices andE edges., witch is, in fact, a “super-saturated” version of a wellknown theorem of Erdős and Gallai.

On weighted multiway cuts in trees

A min—max theorem is developed for the multiway cut problem of edge-weighted trees. We present a polynomial time algorithm to construct an optimal dual solution, if edge weights come in unary representation. Applications to biology also require some more complex edge weights. We describe a dynamic programming type algorithm for this more general problem from biology and show that our min—max theorem does not apply to it.

Degree-based graph construction

Degree-based graph construction is a ubiquitous problem in network modelling (Newman et al 2006 The Structure and Dynamics of Networks (Princeton Studies in Complexity) (Princeton, NJ: Princeton University Press), Boccaletti et al 2006 Phys. Rep. 424 175), ranging from social sciences to chemical compounds and biochemical reaction networks in the cell. This problem includes existence, enumeration, exhaustive construction and sampling questions with aspects that are still open today. Here we give necessary and sufficient conditions for a sequence of nonnegative integers to be realized as a simple graphs degree sequence, such that a given (but otherwise arbitrary) set of connections from an arbitrarily given node is avoided. We then use this result to present a swap-free algorithm that builds all simple graphs realizing a given degree sequence. In a wider context, we show that our result provides a greedy construction method to build all the f-factor subgraphs (Tutte 1952 Can. J. Math. 4 314) embedded within Kn Sk, where Kn is the complete graph and Sk is a star graph centred on one of the nodes.

Covering a graph by complete bipartite graphs

We prove the following theorem: the edge set of every graph G on n vertices can be partitioned into the disjoint union of complete bipartite graphs such that each vertex is contained by at most c(n/log n) of the bipartite graphs.

On Ramsey—Turán type theorems for hypergraphs

AbstractLetHr be anr-uniform hypergraph. Letg=g(n;Hr) be the minimal integer so that anyr-uniform hypergraph onn vertices and more thang edges contains a subgraph isomorphic toHr. Lete =f(n;Hr,εn) denote the minimal integer such that everyr-uniform hypergraph onn vertices with more thane edges and with no independent set ofεn vertices contains a subgraph isomorphic toHr.We show that ifr>2 andHr is e.g. a complete graph then

Intersecting sperner families and their convex hulls

\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r )

Extremal hypergraph problems and convex hulls

while for someHr with