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| 2. |
- Forbes, Anthony D., et al.
(författare)
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Existence results for pentagonal geometries
- 2022
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Ingår i: The Australasian Journal of Combinatorics. - : University of Queensland Press. - 1034-4942. ; 82:1, s. 95-114
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Tidskriftsartikel (refereegranskat)abstract
- New results on pentagonal geometries PENT(k, r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3, r) systems with the maximum number of opposite line pairs as well as those without any opposite line pairs. A wide-ranging result about PENT(3, r) with any number of opposite line pairs is proved. We also determine the existence spectrum of PENT(4, r) systems with eleven possible exceptions.
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| 3. |
- Izquierdo, Milagros, 1963-, et al.
(författare)
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Isometric Point-Circle Configurations on Surfaces from Uniform Maps
- 2016
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer-Verlag New York. - 2194-1009. ; 159, s. 201-212, s. 201-212
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Tidskriftsartikel (refereegranskat)abstract
- We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with a maximum number of automorphisms for their genus, and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle v4'>v4v4 configurations on p-gonal surfaces with two p-gonal morphisms is given. The image of these configurations on the sphere under the two p-gonal morphisms is also described.
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| 4. |
- Lundqvist, Signe, et al.
(författare)
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Applying the pebble game algorithm to rod configurations
- 2023
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Ingår i: EuroCG 2023.
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Konferensbidrag (refereegranskat)abstract
- We present results on rigidity of structures of rigid rods connected in joints: rod configurations. The underlying combinatorial structure of a rod configuration is an incidence structure. Our aim is to find simple ways of determining which rod configurations admit non-trivial motions, using the underlying incidence structure.Rigidity of graphs in the plane is well understood. Indeed, there is a polynomial time algorithm for deciding whether most realisations of a graph are rigid. One of the results presented here equates rigidity of sufficiently generic rod configurations to rigidity of a related graph. As a consequence, itis possible to determine the rigidity of rod configurations using the previously mentioned polynomial time algorithm. We use this to show that all v3-configurations on up to 15 points and all triangle-free v3-configurations on up to 20 points are rigid in regular position, if such a realisation exists. We also conjecture that the smallest v3-configuration that is flexible in regular position is a previously known 283-configuration.
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| 5. |
- Bras-Amorós, Maria, et al.
(författare)
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The semigroup of combinatorial configurations
- 2012
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Ingår i: Semigroup Forum. - : Springer. - 0037-1912 .- 1432-2137. ; 84:1, s. 91-96
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Tidskriftsartikel (refereegranskat)abstract
- We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.
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| 6. |
- Conder, Marston, et al.
(författare)
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New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces
- 2019
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Ingår i: Ars Mathematica Contemporanea. - : Drustvo Matematikov, Fizikov in Astronomov,Society of Mathematicians, Physicists and Astronomers. - 1855-3966 .- 1855-3974. ; 17:1, s. 1-35
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Tidskriftsartikel (refereegranskat)abstract
- The question of how to find the smallest genus of all embeddings of a given finite connected graph on an orientable (or non-orientable) surface has a long and interesting history. In this paper we introduce four new approaches to help answer this question, in both the orientable and non-orientable cases. One approach involves taking orbits of subgroups of the automorphism group on cycles of particular lengths in the graph as candidates for subsets of the faces of an embedding. Another uses properties of an auxiliary graph defined in terms of compatibility of these cycles. We also present two methods that make use of integer linear programming, to help determine bounds for the minimum genus, and to find minimum genus embeddings. This work was motivated by the problem of finding the minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that problem but also in answering several other open questions.
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| 7. |
- Stokes, Klara, et al.
(författare)
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Associating a numerical semigroup to the triangle-free configurations
- 2011
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Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346 .- 1930-5338. ; 5:2, s. 351-371
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Tidskriftsartikel (refereegranskat)abstract
- It is proved that a numerical semigroup can be associated to the triangle-free -configurations, and some results on existence are deduced. For example it is proved that for any there exists infinitely many -configurations. Most proofs are given from a graph theoretical point of view, in the sense that the configurations are represented by their incidence graphs. An application to private information retrieval is described.
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