We consider an extremal problem motivated by a paper of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge-colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given r ≥ t ≥ 2, we look for n-vertex graphs that admit the maximum number of r-edge-colorings such that at...

According to Paul Erdős [Some notes on Turán’s mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Turán who “created the area of extremal problems in graph theory”. However, without a doubt, Paul Erdős popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and al...

A cancellative hypergraph has no three edges A;B;C with ADBCC: We give a new short proof of an old result of Bollobás, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3-graph. One of the two forbidden subhypergraphs in a cancellative 3-graph is F5 1⁄4 fabc; abd; cdeg: For nX33 we show that the maximum number of triples on n vert...

Let f(n, p) be the maximum number of edges in a graph on n vertices with p perfect matchings. Dudek and Schmitt proved that there exist constants np and cp so that for even n ≥ np, f(n, p) = n 2 4 + cp; we call a graph p-extremal if it has p perfect matchings and n2 4 + cp edges. In this paper, we develop structural theorems in matching theory to study p-extremal graphs and use them in a new co...

For any xed graph F , we say that a graph G is F -free if it does not contain F as a subgraph. We denote by ex(n, F ) the maximum number of edges in a n-vertex graph which is F -free, known as the Turán number of F . In 1974, Erd®s and Rothschild considered a related question where we count the number of certain colorings. Given an integer r, by an r-coloring of a graph G we mean any r-edgecolo...

The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements x such that the image of (ad x) lies in the subspace spanned by x. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of ext...

The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called extremal optimization (EO), is compared with simulated annealing (SA) in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult...

To determine the mechanism of molecular evolution, a reconciliation graph is constructed from two heterogeneous trees, which are referred to as ordered trees. In the reconciliation graph, the leaf nodes of the two ordered trees face each other. Furthermore, leaf nodes with the same label name are connected to each other by an edge. To carry out reconciliation work efficiently, it is necessary t...

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. Its considerable applications are found in a variety of fields. In this paper, we determine the maximum value of Kirchhoff index among the unicyclic graphs with fixed number of vertices and maximum degree, and characterize the corresponding extremal graph.

Szemerédi’s Regularity Lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi’s Lemma can be thought of as a result in analysis. We show three different analytic interpretations.