1. Extremal totally regular mixed graphs and partially oriented incidence graphs of projective and biaffine planesTatiana Bagin Jajcay, Robert Jajcay, György Kiss, István Porupsánszki, 2025, original scientific article Abstract: An (r, z; g)-mixed graph is a graph containing both edges and darts satisfying the regularity property that each vertex of the graph is incident to r edges, z ingoing and z outgoing darts (called total regularity), and being of oriented girth g, i.e., containing an oriented cycle of length g, and no shorter oriented cycles. The problem addressed in this paper is analogous to the Cage Problem and calls for determining the orders of the smallest totally regular (r, z; g)-mixed graphs. We derive several upper and lower bounds on the orders of such minimal graphs, study the relations between these extremal graphs and their non-oriented or digraphical counterparts, and focus on properties of totally regular mixed graphs obtained by replacing some of the edges of the incidence graphs of projective and biaffine planes by darts. We also introduce two constructions based on introducing additional edges or darts into induced subgraphs of these incidence graphs.
Keywords: totally regular mixed graph, girth, projective plane, biaffine plane
Published in RUP: 04.06.2026; Views: 114; Downloads: 7
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2. On optimal λ-separable packings in the planeKároly Bezdek, Zsolt Lángi, 2025, original scientific article Abstract: Let P be a packing of circular disks of radius ρ > 0 in the Euclidean, spherical, or hyperbolic plane. Let 0 ≤ λ ≤ ρ. We say that P is a λ-separable packing of circular disks of radius ρ if the family P′ of disks concentric to the disks of P having radius λ form a totally separable packing, i.e., any two disks of P′ can be separated by a line which is disjoint from the interior of every disk of F′. This notion bridges packings of circular disks of radius ρ (with λ = 0) and totally separable packings of circular disks of radius ρ (with λ = ρ). In this note we extend several theorems on the density, tightness, and contact numbers of disk packings and totally separable disk packings to λ-separable packings of circular disks of radius ρ in the Euclidean, spherical, and hyperbolic plane. In particular, our upper bounds (resp., lower bounds) for the density (resp., tightness) of λ-separable packings of unit disks in the Euclidean plane are sharp for all 0 ≤ λ ≤ 1 with the extremal values achieved by λ-separable lattice packings of unit disks. On the other hand, the bounds of similar results in the spherical and hyperbolic planes are not sharp for all 0 ≤ λ ≤ ρ although they do not seem to be far from the relevant optimal bounds either. The proofs use local analytic and elementary geometry and are based on the so-called refined Molnár decomposition, which is obtained from the underlying Delaunay decomposition and as such might be of independent interest.
Keywords: Euclidean, spherical and hyperbolic plane, λ-separable packing, density, tightness, contact number, refined Molnar decomposition
Published in RUP: 21.10.2025; Views: 696; Downloads: 9
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3. Partial geometries with regular automorphism groups : master’s thesisAdisa Hodžić, 2024, master's thesis Keywords: (near-) linear space, projective plane, affine plane, partial geometry, generalized quadrangle, strongly regular graph, partial difference set, automorphism group
Published in RUP: 25.12.2024; Views: 2373; Downloads: 54
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5. Combinatorial configurations, quasiline arrangements, and systems of curves on surfacesJürgen Bokowski, Jurij Kovič, Tomaž Pisanski, Arjana Žitnik, 2018, original scientific article Keywords: pseudoline arrangement, quasiline arrangement, projective plane, incidence structure, combinatorial configuration, topological configuration, geometric configuration, sweep, wiring diagram, allowable sequence of permutations, maps on surfaces
Published in RUP: 03.01.2022; Views: 2473; Downloads: 32
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