Discrete Mathematics

The team currently includes four leader-focused groups. In 2023, J. Przybyło, using probabilistic methods, continued to investigate the Faudree-Lehel conjecture and started a collaboration with two PhD students. The largest group of M. Woźniak (with 3 PhD students) continued to focus on graph colouring, working in the context of distinguishing certain graph structures by labeling vertices and arcs with group elements, and breaking automorphisms. A. Żak with his two PhD students investigated hypergraphs and lattice saturation as practical models of computer systems. M. Pilśniak's group researched infinite graphs and continued her grant on domination problems in graphs, which culminated in the defence of the doctoral thesis of one of her doctoral students. The aim of the team's research is therefore to produce new results in general discrete mathematics, with a particular focus on those areas where team members already have significant achievements, but also to initiate interesting research in new topics. For example, we have a publication with a group of mathematicians from the University of Ulm, which initiates research on the new concept of edge majority colourings of graphs. In 2023, we made important progress on the well-known Faudree-Lehel conjecture on the irregularity strength of d-regular graphs, which has been open for more than 35 years. Namely, we were able to prove an asymptotic version of this conjecture for the full spectrum of the values of d. Furthermore, we have shown a literal version of the conjecture for sufficiently dense d-regular graphs. We were also able to obtain analogous results for a more difficult, generalised version of this conjecture, where arbitrary graphs, not just regular ones, are analysed in the context of their minimum degree, instead of d. In another paper, we dealt with generalisations of so-called piercing sequences, which have their origin in a question formulated in the 1950s by Steinhaus. Here we obtained a number of results, in particular we were able to significantly improve on the previously best constraints obtained by Konyagin. We also considered arc colourings of symmetric digraphs breaking all non-trivial automorphisms. General colourings and proper colourings with respect to various definitions of neighbouring arcs were investigated. Together with two other papers, this publication fully solves the problem of optimal bounds on the minimum numbers of colours in all types of proper colourings of arcs breaking automorphisms of symmetric digraphs. In doing so, some interesting hypotheses were raised. List colourings which break automorphisms were continued for infinite graphs. The conjecture concerning a certain problem of existence of Hamiltonian cycles in homogeneous hypergraphs, posed in a full version by the authors in 2013 and earlier in a partial version by G. Y. Katona, was also partially solved (but for a significant range of the parameter). At the same time, results with potential applications in theoretical computer science concerning the integrity of lattice graphs, which are popular structures of computer architectures were achieved. An asymptotic result was obtained for the problem posed in [Bagga et al., Discrete Appl. Math. 1992] concerning the integrity of plane lattices. On the other hand, in another paper, a result about the division of an abelian 2-Sylow simple group into an arbitrary abelian subgroups was generalised. Several applications of this result to magic and antimagic labelling of graphs were also presented. Overall, the research intensifies in all four threads around open conjectures and work with doctoral students. Publication activity remains consistently high, ensuring professional promotions within generally accepted time standards. PhDs are finishing on schedule, one person is preparing for a habilitation application, two more for a professorship.

A distinctive feature of the team is the high internationalisation of research. Collaborations with mathematicians from foreign universities have resulted in dozens of joint publications in prestigious journals in discrete mathematics between 2020 and 2023 alone. These are mainly from universities in Bordeaux and Orsay (France), Duluth, Auburn and Hamilton (USA), Leoben and Graz (Austria), Kosice (Slovakia), Johannesburg (South Africa), Freiberg (Germany), Hamilton (Canada), Maribor and Koper (Slovenia), Ostrava (Czech Republic), Messina (Italy), Meshida (Iran) and Istanbul (Turkey).