136 research outputs found
On the split structure of lifted groups
Let ā«ā« be a regular covering projection of connected graphs with the group of covering transformations ā«ā« being abelian. Assuming that a group of automorphisms ā«ā« lifts along to a group ā«ā«, the problem whether the corresponding exact sequence ā«ā« 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 ā«ā« nor the action ā«ā« nor a 2-cocycle ā«ā«, are given. Explicitly constructing the cover ā«ā« together with ā«ā« and ā«ā« as permutation groups on ā«ā« is time and space consuming whenever ā«ā« is largethus, 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 ā«ā« is elementary abelian.Naj bo ā«ā« regularna krovna projekcija povezanih grafov, grupa krovnih transformacij ā«ā« pa naj bo abelova. Ob predpostavki, da se grupa avtomorfizmov ā«ā« dvigne vzdolž ā«ā« do grupe ā«ā«, podrobno analiziramo problem, ali se ustrezno eksaktno zaporedje ā«ā« razcepi glede na Cayleyevo dodelitev napetosti, ki rekonstruira projekcijo do ekvivalence natanÄno. V gornjem kombinatoriÄnem sestavu je razÅ”iritev podana samo implicitno: podani niso ne ā«ā« ne delovanje ā«ā« ne 2-kocikel ā«ā«. Eksplicitno konstruiranje krova ā«ā« ter ā«ā« in ā«ā« kot permutacijskih grup na ā«ā« je Äasovno in prostorsko zahtevno vselej, kadar je ā«ā« veliktako je uporaba implementiranih algoritmov (na primer, HasComplement v Magmi) vse prej kot optimalna. Namesto tega pokažemo, da lahko najnujnejÅ”o informacijo o delovanju in 2-kociklu uÄinkovito izluÅ”Äimo neposredno iz napetosti (ne da bi eksplicitno konstruirali krov in dvignjeno grupo)zdaj bi bilo mogoÄe uporabiti standardno metodo reduciranja problema na reÅ”evanje sistema linearnih enaÄb nad celimi Å”tevili. Vendar tukaj uberemo malce drugaÄen pristop, ki sploh ne zahteva nobenega poznavanja kohomologije. Äasovno in prostorsko zahtevnost formalno analiziramo za vse primere, ko je ā«ā« elementarna abelova
Regular maps with nilpotent automorphism groups
AbstractWe study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps HĆK, where Aut(H) is a 2-group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2
The strongly distance-balanced property of the generalized Petersen graphs
A graph ā«ā« is said to be strongly distance-balanced whenever for any edge ā«ā« of ā«ā« and any positive integer ā«ā«, the number of vertices at distance ā«ā« from ā«ā« and at distance ā«ā« from ā«ā« is equal to the number of vertices at distance ā«ā« from ā«ā« and at distance ā«ā« from ā«ā«. It is proven that for any integers ā«ā« and ā«ā«, the generalized Petersen graph GPā«ā« is not strongly distance-balanced
Semiregular automorphisms of vertex-transitive graphs of certain valencies
AbstractIt is shown that a vertex-transitive graph of valency p+1, p a prime, admitting a transitive action of a {2,p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. MaruÅ”iÄ, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69ā81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605ā615])
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose
contraction produces another triangulation of the surface. We prove that every
irreducible triangulation of a surface with Euler genus has at most
vertices. The best previous bound was .Comment: v2: Referees' comments incorporate
Quotients of incidence geometries
We develop a theory for quotients of geometries and obtain sufficient
conditions for the quotient of a geometry to be a geometry. These conditions
are compared with earlier work on quotients, in particular by Pasini and Tits.
We also explore geometric properties such as connectivity, firmness and
transitivity conditions to determine when they are preserved under the
quotienting operation. We show that the class of coset pregeometries, which
contains all flag-transitive geometries, is closed under an appropriate
quotienting operation.Comment: 26 pages, 5 figure
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