1. Regular colouring defect of a cubic graph and the conjectures of Fan-Raspaud and FulkersonJán Karabáš, Edita Máčajová, Roman Nedela, Martin Škoviera, 2026, original scientific article Abstract: We introduce a new invariant of a cubic graph – its regular colouring defect – which is defined as the smallest number of edges left uncovered by any collection of three perfect matchings that have no edge in common. This invariant is a modification of colouring defect, an invariant introduced by Steffen in 2025, whose definition does not require the empty intersection condition. In this paper we discuss the relationship of this invariant to the well-known conjectures of Fulkerson (1971) and Fan and Raspaud (1994) and prove that colouring defect and regular colouring defect can be arbitrarily far apart.
Keywords: cubic graph, perfect matching, colouring defect, Fulkerson Conjecture, Fan and Raspaud Conjecture
Published in RUP: 03.03.2026; Views: 582; Downloads: 18
 Full text (313,05 KB) |
2. |
3. |
4. Regular embeddings of cycles with multiple edges revisitedKan Hu, Roman Nedela, Martin Škoviera, Naer Wang, 2015, original scientific article Abstract: Regularne vložitve ciklov z večkratnimi povezavami se pojavljajo v literaturi že kar nekaj časa, tako v topološki teoriji grafov kot tudi izven nje. Ta članek izriše kompletno podobo teh zemljevidov na ta način, da povsem opiše, klasificira in enumerira regularne vložitve ciklov z večkratnimi povezavami tako na orientabilnih kot tudi na neorientabilnih ploskvah. Večina rezultatov je sicer znana v tej ali oni obliki, toda tu so predstavljeni iz poenotenega zornega kota, osnovanega na teoriji končnih grup. Naš pristop daje dodatno informacijo tako o zemljevidih kot o njihovih grupah avtomorfizmov, priskrbi pa tudi dodaten vpogled v njihove odnose.
Keywords: regularna vložitev, večkratna povezava, Hölderjev izrek, Möbiusov zemljevid, regular embedding, multiple edge, Hölder's Theorem, Möbius map
Published in RUP: 15.10.2015; Views: 5616; Downloads: 115
 Link to full text |
5. Maximum genus, connectivity, and Nebeský's theoremDan Steven Archdeacon, Michal Kotrbčík, Roman Nedela, Martin Škoviera, 2015, original scientific article Keywords: maksimalen rod, Nebeskýnov rod, Bettijevo število, povezanost, maximum genus, Nebeský theorem, Betti number, connectivity
Published in RUP: 15.10.2015; Views: 4630; Downloads: 156
Link to full text |
6. |
7. |
8. |
9. |
10. Decomposition of skew-morphisms of cyclic groupsIstván Kovács, Roman Nedela, 2011, original scientific article Abstract: A skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity such that for every ▫$x, y \in H$▫ there exists an integer ▫$k$▫ such that ▫$\sigma (xy) = \sigma (x)\sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$\mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, \varphi (n_{2})) = (\varphi (n_{1}), n_{2}) = 1$▫ (▫$\varphi$▫ denotes Euler's function) then all skew-morphisms ▫$\sigma$▫ of ▫$\mathbb Z_n$▫ are obtained as ▫$\sigma = \sigma_1 \times \sigma_2$▫, where ▫$\sigma_i$▫ are skew-morphisms of ▫$\mathbb Z_{n_i}, \; i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$\mathbb Z_n$▫ are automorphisms of ▫$\mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, \varphi(n)) = 1$▫.
Keywords: cyclic group, permutation group, skew-morphism, Schur ring
Published in RUP: 15.10.2013; Views: 6806; Downloads: 114
Link to full text |
