mycoreobjectrosdok_document_0000027969rosdok_document_0000027969rosdokdocumentrosdok_derivate_00002299082026-10-14T12:03:42.560Z2026-10-14T11:51:31.906ZMCRJANITOReditorKEpublished1rosdok_derivate_0000229908truetruederivate_types:fulltextVolltextfulltextwf_edit_epub wf_register_epubstate:publishedstate:publishedveröffentlichtpublishedrosdok_id00005832198676804XOau2026-10-142026-10-14T12:03:39ZrdaConverted from PICA to MODS using Pica2MODS XSLT Transformer 2.14-SNAPSHOT [SCM: "" "" ""] with mode 'DEFAULT'.WorkingpaperOrthogonal covers by multiplication graphsLet K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian.Hans-DietrichGronau1951 -VerfasserInaut138612498MarkusSchmidmeier1963 -VerfasserInaut1730402920000-0003-3365-6666https://purl.uni-rostock.de/rosdok/id00005832urn:nbn:de:gbv:28-rosdok_id00005832-510.18453/rosdok_id00005832510 MathematikMathematisch-Naturwissenschaftliche Fakultätalle Rechte vorbehaltenRechteinhaber Universität Rostockfrei zugänglich (Open Access)Lizenz Metadaten: CC0en1 Online-Ressource (15 Seiten)2007University of Rostock, Institute of MathematicsRostockmonographic20072026Universitätsbibliothek RostockRostock2026Universitätsbibliothek Rostockhttps://purl.uni-rostock.de/rosdok/id00005832Hans-Dietrich O.F. Gronau and Markus Schmidmeierrosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-807/10007-0001doctype:epubdoctype:epubDokumenttypDocument typedoctype:epub.workingpaperdoctype:epub.workingpaperWorkingpaperworking paperdiniPublType:workingPaper diniPublType2022:WorkingPaperinfo:eu-repo/semantics/workingPaperdocumentdiniPublType2022:WorkingPaperdiniPublType2022:WorkingPaperArbeitspapierWorking paperKDSF (Pu37)SDNB:510SDNB:510510 Mathematik510 Mathematicsinstitution:unirostockinstitution:unirostockUniversität RostockUniversity of RostockUniversität RostockUniversität RostockUni.Rostockhttp://d-nb.info/gnd/38329-6institution:unirostock.mnfinstitution:unirostock.mnfMathematisch-Naturwissenschaftliche FakultätFaculty of Mathematics and Natural SciencesUniversität Rostock. Mathematisch-Naturwissenschaftliche FakultätMathematisch-Natur-<br />wissenschaftliche FakultätUni.Rostock.Fakultaet.MNFhttp://d-nb.info/gnd/2147083-2licenseinfo:worklicenseinfo:workWerkworklicenseinfo:work.rightsreservedlicenseinfo:work.rightsreservedalle Rechte vorbehaltenall rights reserved/creativecommons/r/reserved/0.9/88x31.png[DE-28]Urheberrechtsschutz 1.0$gRights Statements$uhttp://rightsstatements.org/vocab/InC/1.0/http://rightsstatements.org/vocab/InC/1.0/http://rightsstatements.org/vocab/InC/1.0/licenseinfo:depositlicenseinfo:depositVeröffentlichungsgenehmigungpermission to storelicenseinfo:deposit.rightsunirostocklicenseinfo:deposit.rightsunirostockRechteinhaber Universität Rostockrightsholder Rostock Universityaccesscondition:openaccessaccesscondition:openaccessfrei zugänglich (Open Access)open accesshttp://purl.org/coar/access_right/c_abf2OAfreeinfo:eu-repo/semantics/openAccess[DE-28]Open Access$gControlled Vocabulary for Access Rights$uhttp://purl.org/coar/access_right/c_abf2licenseinfo:metadatalicenseinfo:metadataLizenzen für Metadatenlicenseinfo:metadata.cc0licenseinfo:metadata.cc0Lizenz Metadaten: CC0license metadata: CC0/creativecommons/p/zero/1.0/88x31.pnghttps://creativecommons.org/publicdomain/zero/1.0/rfc5646:enrfc5646:enEnglischEnglishengengOrthogonal covers by multiplication graphsOrthogonal covers by multiplication graphsgnd:138612498gnd:173040292orcid:0000-0003-3365-6666gnd:138612498gnd:173040292orcid:0000-0003-3365-6666rosdok_document_0000027969-d3750448e49gnd:138612498Hans-Dietrich GronauHans-Dietrich Gronaurosdok_document_0000027969-d3750448e63gnd:173040292orcid:0000-0003-3365-6666Markus SchmidmeierMarkus SchmidmeierHans-Dietrich GronauMarkus SchmidmeierHans-Dietrich GronauMarkus SchmidmeierHans-Dietrich GronauMarkus SchmidmeierRostockUniversity of Rostock, Institute of Mathematicsepub.workingpaperhttps://purl.uni-rostock.de/rosdok/id00005832urn:nbn:de:gbv:28-rosdok_id00005832-510.18453/rosdok_id00005832Let K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian.20072007Preprints aus dem Institut für Mathematikrosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-807/10007-0001https://purl.uni-rostock.de/rosdok/id000058163255797-8Hans-Dietrich O.F. Gronau and Markus Schmidmeierepub.workingpaper1 Gronau_covers_2007.pdf 221254 a87686efe83e1863d55405630f637281 rosdok_id00005832198676804XOau2026-10-142026-10-14T12:03:39ZrdaConverted from PICA to MODS using Pica2MODS XSLT Transformer 2.14-SNAPSHOT [SCM: "" "" ""] with mode 'DEFAULT'.WorkingpaperOrthogonal covers by multiplication graphsLet K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian.Hans-DietrichGronau1951 -VerfasserInaut138612498MarkusSchmidmeier1963 -VerfasserInaut1730402920000-0003-3365-6666https://purl.uni-rostock.de/rosdok/id00005832urn:nbn:de:gbv:28-rosdok_id00005832-510.18453/rosdok_id00005832510 MathematikMathematisch-Naturwissenschaftliche Fakultätalle Rechte vorbehaltenRechteinhaber Universität Rostockfrei zugänglich (Open Access)Lizenz Metadaten: CC0en1 Online-Ressource (15 Seiten)2007University of Rostock, Institute of MathematicsRostockmonographic20072026Universitätsbibliothek RostockRostock2026Universitätsbibliothek Rostockhttps://purl.uni-rostock.de/rosdok/id00005832Hans-Dietrich O.F. Gronau and Markus Schmidmeierrosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-807/10007-0001 2026-10-14T11:51:31.906Z 2026-10-14T12:03:42.560Z 2026-10-24T12:03:42.568Z editorKE {"identifier":"rosdok/id00005832","type":"local_id","additional":"","service":"MCRLocalID","created":"2026-10-14T11:51:31.918Z"} {"identifier":"urn:nbn:de:gbv:28-rosdok_id00005832-5","type":"dnbUrn","additional":"","service":"RosDokURN","created":"2026-10-14T11:51:31.927Z"} {"identifier":"https://purl.uni-rostock.de/rosdok/id00005832","type":"purl","additional":"","service":"RosDokPURL","created":"2026-10-14T11:51:31.930Z","registrationStarted":"2026-10-14T12:03:41.225Z"} {"identifier":"10.18453/rosdok_id00005832","type":"doi","additional":"","service":"RosDokDOI","created":"2026-10-14T11:51:31.933Z","registrationStarted":"2026-10-14T12:03:40.991Z","registered":"2026-10-14T12:03:42.558Z"} MCRJANITORfulltextfile/rosdok_document_0000027969/rosdok_derivate_0000229908/Gronau_covers_2007.pdfOrthogonal covers by multiplication graphsOrthogonal covers by multiplication graphsgnd:138612498gnd:173040292orcid:0000-0003-3365-6666gnd:138612498gnd:173040292orcid:0000-0003-3365-6666rosdok_document_0000027969-d3750448e49gnd:138612498Hans-Dietrich GronauHans-Dietrich Gronaurosdok_document_0000027969-d3750448e63gnd:173040292orcid:0000-0003-3365-6666Markus SchmidmeierMarkus SchmidmeierHans-Dietrich GronauMarkus SchmidmeierHans-Dietrich GronauMarkus SchmidmeierHans-Dietrich GronauMarkus SchmidmeierRostockUniversity of Rostock, Institute of Mathematicsepub.workingpaperhttps://purl.uni-rostock.de/rosdok/id00005832urn:nbn:de:gbv:28-rosdok_id00005832-510.18453/rosdok_id00005832Let K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian.20072007Preprints aus dem Institut für Mathematikrosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-807/10007-0001https://purl.uni-rostock.de/rosdok/id000058163255797-8Hans-Dietrich O.F. Gronau and Markus Schmidmeier[xslt]Saxonrosdok_id00005832https://purl.uni-rostock.de/rosdok/id00005832198676804X10.18453/rosdok_id00005832urn:nbn:de:gbv:28-rosdok_id00005832-5Hans-Dietrich Gronau, Markus SchmidmeierGronau Hans-Dietrich, Schmidmeier MarkusOrthogonal covers by multiplication graphsWorkingpaperworking paperUniversity of Rostock, Institute of Mathematics, 2007Let K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau,…rosdok_id000058160007-000107/1Hans-Dietrich GronauMarkus SchmidmeierOrthogonal covers by multiplication graphsPreprints aus dem Institut für MathematikUniversitätsbibliothek Rostockhttps://purl.uni-rostock.de/rosdok/id00005832Hans-DietrichGronau1951 -VerfasserInaut138612498MarkusSchmidmeier1963 -VerfasserInaut1730402920000-0003-3365-6666[purl]https://purl.uni-rostock.de/rosdok/id00005832[urn]urn:nbn:de:gbv:28-rosdok_id00005832-5[doi]10.18453/rosdok_id00005832open_access20072007epoch:21th_centuryrfc5646:endoctype:epub.workingpaperaccesscondition:openaccessSDNB:510institution:unirostock.mnfstate:published