Coloring minimal Cayley graphs.

In 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 c...

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Detalles Bibliográficos
Autores: García Marco, Ignacio, Knauer, Kolja
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2025
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/224639
Acceso en línea:https://hdl.handle.net/2445/224639
Access Level:acceso abierto
Palabra clave:Teoria de grafs
Geometria combinatòria
Graph theory
Combinatorial geometry
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spelling Coloring minimal Cayley graphs.García Marco, IgnacioKnauer, KoljaTeoria de grafsGeometria combinatòriaGraph theoryCombinatorial geometryIn 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we address a related question proposed by Babai in 1978 by constructing graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter can be viewed as a step towards confirming Babai’s 1994 conjecture – a problem that remains open.Elsevier Ltd.2025202520252025info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion9 p.application/pdfhttps://hdl.handle.net/2445/224639Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció del document publicat a: https://doi.org/10.1016/j.ejc.2024.104108European Journal of Combinatorics, 2025https://doi.org/10.1016/j.ejc.2024.104108cc-by-nc (c) García Marco, Ignacio et al., 2025http://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2445/2246392026-05-29T05:05:01Z
dc.title.none.fl_str_mv Coloring minimal Cayley graphs.
title Coloring minimal Cayley graphs.
spellingShingle Coloring minimal Cayley graphs.
García Marco, Ignacio
Teoria de grafs
Geometria combinatòria
Graph theory
Combinatorial geometry
title_short Coloring minimal Cayley graphs.
title_full Coloring minimal Cayley graphs.
title_fullStr Coloring minimal Cayley graphs.
title_full_unstemmed Coloring minimal Cayley graphs.
title_sort Coloring minimal Cayley graphs.
dc.creator.none.fl_str_mv García Marco, Ignacio
Knauer, Kolja
author García Marco, Ignacio
author_facet García Marco, Ignacio
Knauer, Kolja
author_role author
author2 Knauer, Kolja
author2_role author
dc.subject.none.fl_str_mv Teoria de grafs
Geometria combinatòria
Graph theory
Combinatorial geometry
topic Teoria de grafs
Geometria combinatòria
Graph theory
Combinatorial geometry
description In 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we address a related question proposed by Babai in 1978 by constructing graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter can be viewed as a step towards confirming Babai’s 1994 conjecture – a problem that remains open.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/224639
url https://hdl.handle.net/2445/224639
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.1016/j.ejc.2024.104108
European Journal of Combinatorics, 2025
https://doi.org/10.1016/j.ejc.2024.104108
dc.rights.none.fl_str_mv cc-by-nc (c) García Marco, Ignacio et al., 2025
http://creativecommons.org/licenses/by-nc/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by-nc (c) García Marco, Ignacio et al., 2025
http://creativecommons.org/licenses/by-nc/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 9 p.
application/pdf
dc.publisher.none.fl_str_mv Elsevier Ltd.
publisher.none.fl_str_mv Elsevier Ltd.
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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