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The isomorphism classes of hypermaps of a given genus g [less than or equal to] 6 and a given number d of darts are enumerated. The hypermaps of a given genus g are distinguished up to orientation preserving isomorphisms. The obtained results depend on recent progress in counting rooted hypermaps, in particular, by P. Zograf, M. Kazarian, A. Giorgetti, and T. Walsh. These results can be interpreted as an enumeration of conjugacy classes of subgroups in a free Fuchsian group of rank two with a genus restriction. Bibliography: 36 titles., UDC 519.175.3, 525.162.6 1. Introduction An oriented map is a 2-cell decomposition of a closed orientable surface with a fixed global orientation. Oriented hypermaps are generalizations of oriented maps. While [...]

Thomassen's conjecture states that a cubic graph with sufficiently large cyclic connectivity is hamiltonian. Even the following strong conjecture could hold: A cyclically 7-connected cubic graph is hamiltonian, or it is the Coxeter graph. Assuming the conjecture holds true, to prove the hamiltonicity of cubic Cayley graphs it is sufficient to examine cubic Cayley graphs of girth at most 6. Motivated by this, we characterise cubic Cayley graphs of girth at most six and identify few \hard families" of cubic Cayley graphs of small girth for which we are not able to verify whether they are hamiltonian. [ABSTRACT FROM AUTHOR]

A skew morphism of the cyclic additive group Zn is a bijection φ on Zn for which there exists an integer-valued function π : Zn → Z such that φ(0) = 0 and φ(x + y) = φ(x) + φπ(x)(y) for all x; y 2 Zn. A pair of skew morphisms φ: Zn → Zn and eφ: Zm → Zm are reciprocal if (a) the orders of φ and eφ divide m and n, respectively, and (b) the associated power functions π and π are determined by π(x) = eφx(1) and π(y) = φy(1). Reciprocal pairs of skew morphisms of cyclic groups are in one-to-one correspondence with isomorphism classes of regular dessins with complete bipartite underlying graphs. In this paper we determine all reciprocal pairs of skew morphisms of the cyclic groups provided that one of them is a group automorphism. [ABSTRACT FROM AUTHOR]

Background: The research aimed to investigate the incidence of SARS-CoV-2 breakthrough infections and their determinants in a large European cohort of more than 60,000 health workers., Methods: A multicentric retrospective cohort study, involving 12 European centers, was carried out within the ORCHESTRA project, collecting data up to 18 November 2021 on fully vaccinated health workers. The cumulative incidence of SARS-CoV-2 breakthrough infections was investigated with its association with occupational and social-demographic characteristics (age, sex, job title, previous SARS-CoV-2 infection, antibody titer levels, and time from the vaccination course completion)., Results: Among 64,172 health workers from 12 European health centers, 797 breakthrough infections were observed (cumulative incidence of 1.2%). The primary analysis using individual data on 8 out of 12 centers showed that age and previous infection significantly modified breakthrough infection rates. In the meta-analysis of aggregated data from all centers, previous SARS-CoV-2 infection and the standardized antibody titer were inversely related to the risk of breakthrough infection ( p = 0.008 and p = 0.007, respectively)., Conclusion: The inverse correlation of antibody titer with the risk of breakthrough infection supports the evidence that vaccination plays a primary role in infection prevention, especially in health workers. Cellular immunity, previous clinical conditions, and vaccination timing should be further investigated.