The research in the field of discrete mathematics has a long tradition at the Faculty of Sciences, P. J. Šafárik University in Košice (PF UPJŠ in the following text). It was initiated in the second half of the sixties of the last century, when Professor Ernest Jucovič came to PF UPJŠ. In this period he founded the KOšický KOmbinatorický Seminár (KOKOS), which from that time has worked regularly. He gradually invited his students – S. Jendroľ, M. Trenkler, P. Mihók, M. Horňák and later J. Ivančo – to the seminar, and, subsequently, he managed to employ them at PF UPJŠ. The scientific team embarked on solving (at the time) open problems of combinatorial geometry of convex polyhedra and polyhedral maps as well as concerning graph colourings, long cycles in graphs and the structure of hypergraphs. Already the first results were very encouraging and were published in leading world scientific journals. In the following years, the team successfully solved several projects of the state programme of basic research, and, subsequently, became established in Czechoslovak mathematics. Results of solutions appeared in quality international journals.
From the beginning, Professor Jucovič aimed to form international contacts and created a space for his students to have the opportunity to establish a cross-border cooperation. Team members met several world-famous personalities (P. Erdős, V. T. Sós, G. O. H. Katona, L. Fejes-Tóth, H. Sachs, M. Fiedler, J. Sedláček, J. Bosák) as well as younger mathematicians beginning their scientific career (L. Lovász, H. Walther, M. Borowiecki, Z. Skupień). Many of these contacts were later developed into an intensive international co-operation.
Within the framework of KOKOS, several other personalities of discrete mathematics in Košice (M. Bača, J. Bucko, A. Feňovčíková and M. Klešč from TU Košice, J. Ivančo from PF UPJŠ) or of the corporate sphere (R. Vasky of VSL Software) got their expertise.
After the political changes in 1989, a new perspective was opened to the members of the Jucovič group, as the freedom of travelling allowed the sporadic written contacts to be replaced by personal ones. In 1991 Professor Jucovič retired, and the team’s leadership came, in a natural way, to S. Jendroľ. Team members intensified their international scientific co-operation; in this direction, as a key element for the further orientation of the group, one can recognise the collaboration with colleagues from Ilmenau in Germany and the subsequent co-organizing the Workshop Cycles and Colourings since 1992. In 2017, the 26th edition of this event has been held. Thanks to the workshop, it was possible to welcome in Slovakia almost all top experts working on subjects in the sphere of interest of the team. In the last two decades, the members of the group have been regularly invited for stays abroad and to lecture at international conferences; several of them have also become members of editorial boards of scientific journals.
A great attention was also devoted to the education of followers of the heritage of the Jucovič group. Since the mid-nineties, much of the content of all mathematically oriented study programme at PF UPJŠ was devoted to discrete mathematics and its applications. In 2002, at the Faculty of Sciences – as the first in Slovakia – a postgraduate study in the field of Discrete Mathematics was established, in the frame of which, up to now, 23 holders of the PhD degree were supervised.
After 1990, the research team at PF UPJŠ focused on discrete mathematics was completed with successful students of the original members of the team of prof. Jucovič: R. Soták (supervisor M. Horňák), G. Semanišin (supervisor P. Mihók), I. Fabrici and T. Madaras (supervisor S. Jendroľ). The youngest member of the KOSDIM team is another student of S. Jendroľ, M. Maceková.
From 1990, when grant schemes started to operate in Slovakia, members of the present KOSDIM team regularly receive funding to support their research and expand their spectrum of scientific interests. There are systematically studied the global and local structural properties of graphs embedded in a sphere and in the surfaces of higher genera, or geometric representations of graphs in metric and Euclidean spaces, respectively. Also, an intensive exploration is devoted to various colourings and labellings of graphs in general as well as for specific families constrained by different conditions on graph diagrams, as well as problems concerning longest cycles in graphs, partitions and decompositions of graphs. An attention is also paid to applications of graphs, especially in the field of social and complex network analysis.
