1. On simple groups that admit a string C-group representationDimitri Leemans, Adrien Vandenschrick, 2026, original scientific article Abstract: We prove that a simple group admits at least one string C-group representation if and only if it is not one of PSL(3, q), PSU(3, q),
PSL(4, 2^n), PSU(4, 2^n), PSU(4, 3), PSU(5, 2), A₆, A₇, M₁₁, M₂₂, M₂₃ or McL.
Keywords: simple groups, string C-groups
Published in RUP: 05.01.2026; Views: 324; Downloads: 0
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2. Faithful and thin non-polytopal maniplexesDimitri Leemans, Micael Toledo, 2025, original scientific article Abstract: Maniplexes are coloured graphs that generalise maps on surfaces and abstract polytopes. Each maniplex uniquely defines a partially ordered set that encodes information about its structure. When this poset is an abstract polytope, we say that the associated maniplex is polytopal. Maniplexes that have two properties, called faithfulness and thinness, are completely determined by their associated poset, which is often an abstract polytope. We show that all faithful thin maniplexes of rank three are polytopal. So far only one example, of rank four, of a thin maniplex that is not polytopal was known. We construct the first infinite family of maniplexes that are faithful and thin but are non-polytopal for all ranks greater than three.
Keywords: maniplex, polytope, faithful, thin, regular cover
Published in RUP: 18.09.2025; Views: 520; Downloads: 9
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3. A note on girth-diameter cagesGabriela Araujo-Pardo, Marston D. E. Conder, Natalia García-Colín, György Kiss, Dimitri Leemans, 2025, original scientific article Abstract: In this paper we introduce a problem closely related to the Cage Problem and the Degree Diameter Problem. For integers k ≥ 2, g ≥ 3 and d ≥ 1, we define a (k; g, d)-graph to be a k-regular graph with girth g and diameter d. We denote by n₀(k; g, d) the smallest possible order of such a graph, and, if such a graph exists, we call it a (k; g, d)-cage. In particular, we focus on (k; 5, 4)-graphs. We show that n₀(k; 5, 4) ≥ k² + k + 2 for all k, and report on the determination of all (k; 5, 4)-cages for k = 3, 4 and 5 and of examples with k = 6, and describe some examples of (k; 5, 4)-graphs which prove that n₀(k; 5, 4) ≤ 2k² for infinitely many k.
Keywords: cages, girth, degree-diameter problem
Published in RUP: 10.06.2025; Views: 764; Downloads: 15
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