Raziskovalni matematični seminar - Arhiv

In the last century Geometry underwent several substantial extensions and revisions based on the fundamental revolutions that it lived through in the XIXth century.

The purpose of the lecture is to discuss several aspects of these transformations: the new concepts that emerged from these new points of view, the new perspectives that could be drawn, some bringing together the continuous and discrete viewpoints, some classical problems that could be solved, and the new interactions with other disciplines that went along.

The lecture is meant for a general mathematical audience.

This is a presentation of PhD thesis topic. The PhD thesis will consist of three interrelated parts: 1-perfectly orientable graphs, graph products, and the price of connectivity. The central common theme among the three parts is the study of graph classes. In particular, we will examine three well known intersection graph classes which will play a key role in many of our results: chordal, interval, and circular arc graphs.

Following the terminology of Kammer and Tholey, we say that an orientation of a graph is 1-perfect if the out-neighborhood of every vertex induces a tournament, and that a graph is 1-perfectly orientable (1-p.o. for short) if it has a 1-perfect orientation. 1-perfectly orientable graphs are known to be polynomially recognizable, but a complete structural understanding of the class is still an open question. In the presentation we will discuss characterizations of 1-perfectly orientable graphs in various graph classes, including nontrivial product graphs for each of the four standard graph products (Cartesian, direct, strong, and lexicographic).

For a family of graphs F, an F-transversal of a graph G is a subset S in V(G) that intersects every subset of V(G) that induces a subgraph isomorphic to a graph in F. We denote by t_F(G) be the minimum size of an F-transversal of G, and by ct_F(G) be the minimum size of an F-transversal of G that induces a connected graph. For a class of connected graphs, we say that the price of connectivity of F-transversals is multiplicative if, for all G in the class, ct_F(G)/t_F(G) is bounded by a constant, and additive if ct_F(G)-t_F(G) is bounded by a constant. The price of connectivity is identical if t_F(G) and ct_F(G) are always equal and unbounded if ct_F(G) cannot be bounded in terms of t_F(G). The price of connectivity will be discussed in the context of hereditary graph classes defined by a single forbidden induced subgraph.

A graph G of even order is k-extendable if it has at least 2 k+2 vertices, contains a matching of size k, and if every k-matching is contained in a perfect matching of G. In the talk we will discuss 1-extendability and 2-extendability of Cartesian product of graphs and lexicographic products of graphs.

For a positive integer k, a graph is called a k-circulant if its automorphism group contains a cyclic semiregular subgroup with k orbits on the vertices. In this talk, I am going to discuss the following question: for which k there exist infinitely many cubic arc-transitive k-circulants? This is based on a joint work with Michael Giudici, Cai Heng Li and Gabriel Verret.

Abstract. A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. 
In this talk we will discuss a classification (in progress) of finite simple groups in which subgroups of odd indices are pronormal.