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63. Določeni razredi (hiper)grafov in njihove algebraične lastnosti : doktorska disertacijaPaweł Petecki, 2016, doctoral dissertation Keywords: hypergraph, hamiltonian cycle, decomposition, double generalized Petersen graph, automorphism group, vertex-transitive, sign graph, L-eigenvalue, lollipop graph Published in RUP: 09.08.2016; Views: 3273; Downloads: 31 Link to full text |
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68. On the split structure of lifted groupsAleksander Malnič, Rok Požar, 2016, original scientific article Abstract: Let ▫$\wp \colon \tilde{X} \to X$▫ be a regular covering projection of connected graphs with the group of covering transformations ▫$\rm{CT}_\wp$▫ being abelian. Assuming that a group of automorphisms ▫$G \le \rm{Aut} X$▫ lifts along $\wp$ to a group ▫$\tilde{G} \le \rm{Aut} \tilde{X}$▫, the problem whether the corresponding exact sequence ▫$\rm{id} \to \rm{CT}_\wp \to \tilde{G} \to G \to \rm{id}$▫ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither ▫$\tilde{G}$▫ nor the action ▫$G\to \rm{Aut} \rm{CT}_\wp$▫ nor a 2-cocycle ▫$G \times G \to \rm{CT}_\wp$▫, are given. Explicitly constructing the cover ▫$\tilde{X}$▫ together with ▫$\rm{CT}_\wp$▫ and ▫$\tilde{G}$▫ as permutation groups on ▫$\tilde{X}$▫ is time and space consuming whenever ▫$\rm{CT}_\wp$▫ is large; thus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever ▫$\rm{CT}_\wp$▫ is elementary abelian. Keywords: algorithm, abelian cover, Cayley voltages, covering projection, graph, group extension, group presentation, lifting automorphisms, linear systems over the integers, semidirect product Published in RUP: 15.10.2015; Views: 2864; Downloads: 157 Full text (422,56 KB) |
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