Resolvable configurations
type: Item
instance of: scholarly article
title: Resolvable configurations
zbMATH Open document ID: 1464.51005
author: Gábor Gévay
published in: Discrete Applied Mathematics
publication date: 2021-05-19T00:00:00Z
review text: In the paper under review, the author introduces a new class of configurations called resolvable. An incidence structure \(\mathcal{S}\) is an ordered triple \(\mathcal{S} = (\mathcal{P},\mathcal{B},I)\) of mutually disjoint sets such that \(\mathcal{P}\) is the set of points, \(\mathcal{B}\) is the set of blocks, and \(I \subseteq \mathcal{P} \times \mathcal{B}\) is an incidence relation. Let \(v\), \(b\), \(r\), \(k\), \(\lambda\) be positive integers, and consider the following conditions: (C1) \(|\mathcal{P}| = v\); (C2) \(|\mathcal{B}| = b\); (C3) each point is incident with precisely \(r\) blocks; (C4) each block is incident with precisely \(k\) points; (C5) two distinct points are incident with at most one block; (C6) every pair of distinct points is incident with precisely \(\lambda\) block. A combinatorial configuration \((v_{r},b_{k})\) is an incidence structure satisfying conditions (C1)--(C4). If the condition (C5) is also satisfied, then the configuration is called lineal. For integers \(v > k \geq 2\), an incidence structure is called a \((v,k,\lambda )\)-balanced incomplete block design if it satisfies conditions (C1)--(C4) and (C6). This name is usually abbreviated to \((v,k,\lambda)\)-BIBD. A geometric configuration is, in the simplest case, an incidence structure in which each \(\mathcal{P}\) and \(\mathcal{B}\) consist of points and (straight) lines, respectively, in the Euclidean or real projective plane, such that the conditions (C1)--(C4) are satisfied. Every geometric configuration of type \((v_{r},b_{k})\) has an underlying combinatorial configuration of the same type. On the other hand, not every combinatorial configuration can be realized as a geometric configuration. Now suppose that \(\mathcal{R} = (\mathcal{P}, \mathcal{B})\) is a \((v,k,\lambda)\)-BIBD with set of points \(\mathcal{P}\) and set of blocks \(\mathcal{B}\). A parallel class in \(\mathcal{R}\) is a set of blocks such that each element of \(\mathcal{P}\) is incident with pre…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 51A20
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 7349670
zbMATH Keywords: geometric configuration
zbMATH Keywords: point-circle configuration
zbMATH Keywords: resolvable block design
zbMATH Keywords: resolution scheme
MaRDI profile type: Publication
full work available at URL: j.dam.2019.02.019
OpenAlex ID: W4210298348
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Wikidata QID: Q128288754
DOI: 10.1016/J.DAM.2019.02.019
description: scientific article; zbMATH DE number 7349670