matematika (Md)
Long cycles in 1-planar graphs
The motivation for the presented topic is the famous Tutte's theorem stating that every 4-connected planar graph is Hamiltonian. A natural extension of the class of planar graphs are the so-called 1-planar graphs (which have a planar diagram in which each edge is crossed by at most one other edge), their maximum possible vertex connectivity is 7. The existence of non-Hamiltonian 5-connected 1-planar graphs is known, as well as the fact that the maximal 7-connected 1-planar graph is Hamiltonian. There is a hypothesis that every 6-connected 1-planar graph is Hamiltonian.
The main task of this topic is to find necessary or sufficient conditions for Hamiltonicity of 1-planar graphs (regarding their connectivity), or to find a guaranteed bound on the length of the longest cycle for graphs that are not Hamiltonian.
I. Fabrici, J. Harant, T. Madaras, S. Mohr, R. Soták, C.T. Zamfirescu, Long cycles and spanning subgraphs of locally maximal 1‐planar graphs. J Graph Theory 95 (2020), 125–137. W.T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956), 99–116.
RNDr. Igor Fabrici, Dr. rer. nat.
matematika (Md)
Clusters of small degree elements in plane graphs
The aim of the project is to study subgraphs in planar graphs, which are determined in terms of vertices or edges (or, possibly, by both types of given elements) of small degrees. Various existing results in this area are the subject of research in the so-called light graphs theory, developed in more detail since 1995 by the Košice school of discrete mathematics; in further investigation of this topic, we will focus on dual analogues of statements on light paths in polyhedral graphs based on the concept of chains of small faces without self-touchings, or induced light paths/subgraphs, as well as obtaining new results on related structurally motivated invariants of subgraphs in polyhedral graphs (e.g. the gravity of a graph in a class of graphs).
The aim of the project is to study subgraphs in planar graphs, which are determined in terms of vertices or edges (or, possibly, by both types of given elements) of small degrees. Various existing results in this area are the subject of research in the so-called light graphs theory, developed in more detail since 1995 by the Košice school of discrete mathematics; in further investigation of this topic, we will focus on dual analogues of statements on light paths in polyhedral graphs based on the concept of chains of small faces without self-touchings, or induced light paths/subgraphs, as well as obtaining new results on related structurally motivated invariants of subgraphs in polyhedral graphs (e.g. the gravity of a graph in a class of graphs).
S. Jendroľ, H.-J- Voss: Light subgraphs of graphs embedded in the plane—A survey, Discrete Mathematics 313(4) (2013) 406-421 O.V. Borodin, A.O. Ivanova: New results about the structure of plane graphs: A survey, AIP Conference Proceedings 1907 (2017) 030051 other journal publications electronic information resources
prof. RNDr. Tomáš Madaras, PhD.
matematika (Md)
Community structure of complex networks
The aim of the project is to study and examine various approaches to the detection of cohesive subgroups and communities in graph models of social or complex networks from the perspective of theoretical mathematics, as well as algorithmic complexity and limits of their use, with the perspective of generalizing or designing analogues of existing concepts of local communities (such as cliques, n-cliques, n-clans, n-clubs, k-plexes, LS-sets) and developing new algorithms/heuristics for the global community structure of a network given by the decomposition or covering of the set of actors using a system of locally cohesive subgroups.
The aim of the project is to study and examine various approaches to the detection of cohesive subgroups and communities in graph models of social or complex networks from the perspective of theoretical mathematics, as well as algorithmic complexity and limits of their use, with the perspective of generalizing or designing analogues of existing concepts of local communities (such as cliques, n-cliques, n-clans, n-clubs, k-plexes, LS-sets) and developing new algorithms/heuristics for the global community structure of a network given by the decomposition or covering of the set of actors using a system of locally cohesive subgroups.
S. Wassermann, K. Faust: Social Network Analysis: Methods and Applications, Cambridge University Press 1994 U. Brandes, T. Erlebach: Network Analysis. Methodological Foundations, Lecture Notes in Computer Science 3418, Springer-Verlag Berlin Heidelberg 2005
prof. RNDr. Tomáš Madaras, PhD.
matematika (Md)
Random sets and their applications
doc. RNDr. Daniel Klein, PhD.
doc. Mgr. Jozef Kiseľák, PhD.
matematika (Md)
Nonadditive integral operators in the context of uncertainty
The dissertation will focus on the study of nonadditive integral operators with the aim of expanding the theoretical framework and applicability of nonadditive integral methods in the context of uncertainty. Nonadditive integral operators generalize the classical Lebesgue integral and enable more effective modeling of situations where traditional probabilistic approaches are insufficient. The research will concentrate on their application in decision-making processes, information aggregation, and uncertainty modeling. Special attention will be given to the comparison of new and existing types of nonadditive integral operators, their numerical aspects, and optimization possibilities under conditions of imprecise and incomplete data.
1. M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap: Aggregation Functions. Cambridge University Press, Cambridge, 2009. 2. M. Boczek, L. Halčinová, O. Hutník, M. Kaluszka: Novel survival functions based on conditional aggregation operators, Inform. Sci. 580 (2021), 705-719. 3. G. Beliakov, H. Bustince, T. Calvo: A Practical Guide to Averaging Functions, Studies in Fuzziness and Soft Computing, Vol. 329, Springer, 2016.
prof. RNDr. Ondrej Hutník, PhD.
matematika (Md)
From proper to strong edge-coloring of graphs
To study the proper edge coloring of graphs that require stronger conditions for some of the colors. Strong coloring for each color requires that the end vertices of the edges with that color induce matching. It is known that Δ (G) +1 colors are sufficient for regular coloring of the graph G. For strong coloring, it is conjectured that 1.25.Δ(G)^2 colors are sufficient, but currently the best known upper bound is 1,772.Δ(G)^2. Try to find new bounds for some classes of graphs (regular graphs, bipartite graphs, planar graphs, etc.).
N. Gastineau and O. Togni. On S-packing edge-colorings of cubic graphs. Discrete Appl. Math., 259:63–75, 2019 H. Hocquard, D. Lajou, and B. Lužar. Between proper and strong edge-colorings of subcubic graphs. In L. Gasieniec, R. Klasing, and T. Radzik, editors, Combinatorial Algorithms, IWOCA 2020, volume 12126 of Lecture Notes in Comput. Sci., pages 355–367, Springer, 2020.
doc. RNDr. Roman Soták, PhD.
matematika (Md)
Test statistics in elliptical multivariate linear models
Investigate properties and practical applications of tests in multivariate linear models with special variance structures in elliptical distributions.
current journal articles
prof. RNDr. Ivan Žežula, CSc.
matematika (Md)
Test statistics in special multivariate models
Investigate properties and practical applications of tests in multivariate statistical models with special variance structures, especially of those which can be represented as product of beta distributions.
current journal articles
prof. RNDr. Ivan Žežula, CSc.