The scientific and research focus of the team is the study of combinatorial properties of discrete structures, especially the structure and chromatic properties of graphs (plane or embedded into surfaces), the problems of graph labellings and long cycles in graphs. The team is very well composed, both professionally and in the age; its h-index is at least 19.
The leader of the team is prof. RNDr. Stanislav Jendroľ, DrSc. He is student of prof. Jucovič and experienced scientific and pedagogical personality with a wide range of scientific interests. In 2009, was ranked as a top Slovak mathematician by the Slovak academic rating agency ARRA. As a scientist, he wrote (alone or with co-authors) more than 160 original scientific papers published in leading world journals, in which he described a number of new important results on various geometric, topological and combinatorial objects. He significantly contributed to the characterization of the facial and vertex structure of convex polyhedra and polyhedral maps. He described the first known example of a convex polyhedron with non-involutory self-duality (the so-called Jendroľ polyhedron). He discovered (with I. Fabrici, M. Maceková, T. Madaras, M. Tuhársky, R. Soták and foreign colleagues J. Harant, H.-J. Voss, H. Walther) fundamental knowledge about the structure of plane graphs and graphs embedded into orientable surfaces; this led to the foundations of light graph theory, which is currently intensively developed by several research groups around the world. He solved the problem of P. Erdös on the minimum edge weight in the graphs with the prescribed number of vertices and edges (co-author I. Schiermeyer) and together with M. Trenkler, he solved the problem of theoretical chemistry on possible existence of molecules of fulleroids with prescribed geometric structure. He contributed (co-author F. Kardoš) to deeper understanding of the geometric structure of fullerene and fulleroid molecules. He initiated the study of the total labellings of graphs (co-authors M. Bača, M. Miller, J. Ryan) and discovered (with co-authors J. Ivančo, J. Miškuf, R. Soták) numerous properties of the irregular total labellings of graphs. Publications devoted to this topic encountered a vivid response in the world; scientific databases provide more than 85 SCI citations for the principal work of the topic. In cooperation with the co-authors (J. Czap, I. Fabrici, J. Harant, F. Kardoš, R. Soták, E. Škrabuľáková, J. Valiska, M. Voigt, M. Vrbjarová), he the foundations of another viable topic – theory of facial colourings of plane graphs, for which the coloring restrictions concern the graph faces (cyclic, odd, non-repetitive, total, unique-maximum, conflict-free, anagram-free, packing, ranking, WORM, etc.). Until now, he has supervised 17 PhD. (or its equivalent academic titles) students.
Another key member of the team is prof. RNDr. Mirko Horňák, CSc., the student of prof. Jucovič. During post-graduate study, he primarily worked on cell decompositions of orientable surfaces. Later, his scientific interest focused mainly on the chromatic theory of graphs. He obtained significant results on the cyclic chromatic number (co-authors H. Enomoto, S. Jendroľ, J. Zlámalová). He initiated the study of certain chromatic invariants like observability (co-authors J. Černý, R. Soták) and the palette index (co-authors R. Kalinowski, M. Meszka, M. Woźniak). He obtained several results on the achromatic number and achromatic index (co-authors Š. Pčola, J. Puntigán, M. Woźniak). He also dealt with the structural problems of the theory of graphs, namely those ones concerning edge weights in graphs (co-authors A. Gajdoš, P. Hudák, J. Ivančo, S. Jendroľ, T. Madaras, I. Schiermeyer) and decompositions to graphs of prescribed properties (co-authors S. Cichacz, Z. Kocková, A. Marczyk, I. Schiermeyer, M. Woźniak, Zs. Tuza). Until now, he has supervised five PhD. students.
The middle generation in KOSDIM is represented by doc. RNDr. Tomáš Madaras, PhD., doc. RNDr. Gabriel Semanišin, doc. RNDr. Roman Soták, PhD. and RNDr. Igor Fabrici, Dr. rer. nat.
Doc. Madaras is student of prof. Jendroľ. The central theme of his research is the structure of planar and plane graphs or graphs that are geometrically similar to them (co-authors I. Fabrici, D. Hudák, J. Czap, Y. Suzuki), both from the local (the existence of specific small configurations in graphs) as well as from the global point of view (properties of large subgraphs). His other works are devoted to the structure of social or complex networks, namely to the study of mathematical properties of quantitative measures of importance of actors or actor ties within a network (centrality indexes) and detection of communities in the network (co-authors J. Coroničová Hurajová, S. Gago). Part of his research (co-author P. Široczki) also concerns the specific possibilities of graph representations in metric or Euclidean spaces, and graph colourings. Until now, he has supervised three PhD. Students.
Doc. Semanišin is student of doc. RNDr. Peter Mihók, CSc. Under his supervision, he became an expert in the study of hereditary and additive properties of graphs, which naturally generalize the vertex and edge colourings of general graphs (co-authors M. Borowiecki, I. Broere, P. Mihók, R. Vasky, A. Farrugia, B. Richter). He is very successful in applications of these generalized colourings in algorithmic graph theory. Recently, he studies the minimal path vertex covering (co-authors B. Brešar, M. Jakovac, F. Kardoš, J. Katrenič, A. Taranenko), that provides a generalisation of minimum vertex cover, and generalised scheduling problems motivated by various aspects of communications in networks (co-authors F. Galčík, J. Katrenič).
Doc. Soták is student of prof. Horňák. He is creative and active member of team, with a wide scientific range. Together with M. Horňák, he explored several types of edge colouring graphs that distinguish the vertices by colour sets, and later, with his PhD. student J. Rudašová he obtained results for a specific variant of the above colouring which assumes the uniform use of all colors. Many results in the chromatic theory of graphs (concerning colourings forbidding certain bichromatic subgraphs) were published in co-operation with another PhD. student M. Mockovčiaková, colleagues from Slovenia (B. Lužar, R. Škrekovski, J. Kranjc) and other co-authors (F. Kardoš, D. Hudák, Ľ. Bezegová). He participated in generalization of fractional and circular colourings (also their total version) for hereditary and additive properties (co-authors P. Mihók, A. Kemnitz, M. Marangio, G. Karafová). His experiences with distance graphs (which he studied already during his postgraduate studies) were later used in the study of long circles in these graphs (co-authors D. Rautenbach and C. Löwenstein). Until now, he has supervised four PhD. students.
RNDr. Fabrici is student of prof. Jendroľ and prof. Hansjoachim Walther (TU Ilmenau, Germany). The topics of his research are mainly local structural properties (the existence of light subgraphs) of planar and k-planar graphs, colourings of plane graphs and long cycles in graphs. He proved valuable results on facially-constrained colourings of plane graphs (co-authors F. Göring, S. Jendroľ, M. Voigt, M. Vrbjarová). Valuable are also his recent findings on the longest cycles in essentially 4-connected plane graphs (co-authors J. Harant, S. Jendroľ, S. Mohr, J. M. Schmidt). As a co-supervisor, he participated in the supervision of one PhD. student.
RNDr. Mária Maceková, PhD. represents the youngest generation in KOSDIM. Her scientific work was successfully started under the guidance of her PhD. supervisor prof. Jendroľ. She obtained knowledge of structure of short paths in sparse plane graphs (co-authors S. Jendroľ, M. Montassier, R. Soták), which were almost immediately cited.
