Circular backbone coloring for graphs without cycles of size four

Given a graph G = (V(G),E(G)) and a subgraph H = (V(H),E(H)) of G, a q-backbone k-coloring of (G,H) is a function φ : V(G) → { 1 , 2 , 3 ,...,k} such that, for every edge uv ∈ E(G) , we have |φ(u) − φ(v)| ≥ 1 and, for every edge uv ∈ E(H) , we have |φ(u) − φ(v)| ≥ q. The q-backbone chromatic number...

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Detalles Bibliográficos
Autor: Cezar, Alexandre Azevedo
Tipo de recurso: tesis de maestría
Estado:Versión publicada
Fecha de publicación:2016
País:Brasil
Institución:Universidade Federal do Ceará (UFC)
Repositorio:Repositório Institucional da Universidade Federal do Ceará (UFC)
Idioma:inglés
OAI Identifier:oai:repositorio.ufc.br:riufc/75456
Acceso en línea:http://repositorio.ufc.br/handle/riufc/75456
Access Level:acceso abierto
Palabra clave:CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA::MATEMATICA DISCRETA E COMBINATORIA
coloração de grafos
número cromático
coloração backbone circular
grafos planares sem C4
árvore como backbone
graph coloring
chromatic number
circular backbone coloring
planar graphs without C4
tree backbone
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repository_id_str
dc.title.pt_BR.fl_str_mv Circular backbone coloring for graphs without cycles of size four
dc.title.en.pt_BR.fl_str_mv Circular backbone coloring for graphs without cycles of size four
title Circular backbone coloring for graphs without cycles of size four
spellingShingle Circular backbone coloring for graphs without cycles of size four
Cezar, Alexandre Azevedo
CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA::MATEMATICA DISCRETA E COMBINATORIA
coloração de grafos
número cromático
coloração backbone circular
grafos planares sem C4
árvore como backbone
graph coloring
chromatic number
circular backbone coloring
planar graphs without C4
tree backbone
title_short Circular backbone coloring for graphs without cycles of size four
title_full Circular backbone coloring for graphs without cycles of size four
title_fullStr Circular backbone coloring for graphs without cycles of size four
title_full_unstemmed Circular backbone coloring for graphs without cycles of size four
title_sort Circular backbone coloring for graphs without cycles of size four
dc.creator.none.fl_str_mv Cezar, Alexandre Azevedo
author Cezar, Alexandre Azevedo
author_facet Cezar, Alexandre Azevedo
author_role author
dc.contributor.co-advisor.none.fl_str_mv Silva, Ana Shirley Ferreira da
dc.contributor.advisor1.fl_str_mv Araújo, Júlio César Silva
contributor_str_mv Araújo, Júlio César Silva
dc.subject.cnpq.fl_str_mv CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA::MATEMATICA DISCRETA E COMBINATORIA
topic CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA::MATEMATICA DISCRETA E COMBINATORIA
coloração de grafos
número cromático
coloração backbone circular
grafos planares sem C4
árvore como backbone
graph coloring
chromatic number
circular backbone coloring
planar graphs without C4
tree backbone
dc.subject.ptbr.pt_BR.fl_str_mv coloração de grafos
número cromático
coloração backbone circular
grafos planares sem C4
árvore como backbone
dc.subject.en.pt_BR.fl_str_mv graph coloring
chromatic number
circular backbone coloring
planar graphs without C4
tree backbone
description Given a graph G = (V(G),E(G)) and a subgraph H = (V(H),E(H)) of G, a q-backbone k-coloring of (G,H) is a function φ : V(G) → { 1 , 2 , 3 ,...,k} such that, for every edge uv ∈ E(G) , we have |φ(u) − φ(v)| ≥ 1 and, for every edge uv ∈ E(H) , we have |φ(u) − φ(v)| ≥ q. The q-backbone chromatic number of (G,H) , denoted by BBCq (G,H) , is the smallest integer k such that there exists such coloring φ . Similarly, a circular q-backbone k-coloring of (G,H) is a function φ : V(G) → { 1 , 2 , 3 ,...,k} such that, for every edge uv ∈ E(G) , we have |φ(u) − φ(v)| ≥ 1 and, for every edge uv ∈ E(H) , we have k − q ≥ |φ(u) − φ(v)| ≥ q. The circular q-backbone chromatic number of (G,H) , denoted by CBCq (G,H) , is the smallest integer k such that there exists such coloring φ . In this dissertation, we firstly present a brief summary on the results found in literature regarding Backbone Coloring. Then, we prove that if G is a planar graph without cycles of size four and F is a spanning forest of induced paths of G, then CBC2 (G,F) ≤ 7. Lastly, we show the following theorem : if G is a connected graph and k ≥ max {χ(G), χ(G)/ 2 +q} , then there exists a proper k-coloring c of G such that Gc,q is connected, where Gc,q is the subgraph of G such that V(Gc,q ) = V(G) and E(Gc,q ) is the set of edges vw ∈ E(G) that satisfy |c(v)−c(w)| ≥ q.
publishDate 2016
dc.date.issued.fl_str_mv 2016
dc.date.accessioned.fl_str_mv 2023-12-21T16:23:48Z
dc.date.available.fl_str_mv 2023-12-21T16:23:48Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.citation.fl_str_mv CEZAR, Alexandre Azevedo. Circular backbone coloring for graphs without cycles of size four. 2016. 45 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016.
dc.identifier.uri.fl_str_mv http://repositorio.ufc.br/handle/riufc/75456
identifier_str_mv CEZAR, Alexandre Azevedo. Circular backbone coloring for graphs without cycles of size four. 2016. 45 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016.
url http://repositorio.ufc.br/handle/riufc/75456
dc.language.iso.fl_str_mv eng
language eng
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.source.none.fl_str_mv reponame:Repositório Institucional da Universidade Federal do Ceará (UFC)
instname:Universidade Federal do Ceará (UFC)
instacron:UFC
instname_str Universidade Federal do Ceará (UFC)
instacron_str UFC
institution UFC
reponame_str Repositório Institucional da Universidade Federal do Ceará (UFC)
collection Repositório Institucional da Universidade Federal do Ceará (UFC)
bitstream.url.fl_str_mv http://repositorio.ufc.br/bitstream/riufc/75456/1/2016_dis_aacezar.pdf
http://repositorio.ufc.br/bitstream/riufc/75456/2/license.txt
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repository.name.fl_str_mv Repositório Institucional da Universidade Federal do Ceará (UFC) - Universidade Federal do Ceará (UFC)
repository.mail.fl_str_mv bu@ufc.br || repositorio@ufc.br
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spelling Silva, Ana Shirley Ferreira daAraújo, Júlio César Silva2023-12-21T16:23:48Z2023-12-21T16:23:48Z2016CEZAR, Alexandre Azevedo. Circular backbone coloring for graphs without cycles of size four. 2016. 45 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016.http://repositorio.ufc.br/handle/riufc/75456Given a graph G = (V(G),E(G)) and a subgraph H = (V(H),E(H)) of G, a q-backbone k-coloring of (G,H) is a function φ : V(G) → { 1 , 2 , 3 ,...,k} such that, for every edge uv ∈ E(G) , we have |φ(u) − φ(v)| ≥ 1 and, for every edge uv ∈ E(H) , we have |φ(u) − φ(v)| ≥ q. The q-backbone chromatic number of (G,H) , denoted by BBCq (G,H) , is the smallest integer k such that there exists such coloring φ . Similarly, a circular q-backbone k-coloring of (G,H) is a function φ : V(G) → { 1 , 2 , 3 ,...,k} such that, for every edge uv ∈ E(G) , we have |φ(u) − φ(v)| ≥ 1 and, for every edge uv ∈ E(H) , we have k − q ≥ |φ(u) − φ(v)| ≥ q. The circular q-backbone chromatic number of (G,H) , denoted by CBCq (G,H) , is the smallest integer k such that there exists such coloring φ . In this dissertation, we firstly present a brief summary on the results found in literature regarding Backbone Coloring. Then, we prove that if G is a planar graph without cycles of size four and F is a spanning forest of induced paths of G, then CBC2 (G,F) ≤ 7. Lastly, we show the following theorem : if G is a connected graph and k ≥ max {χ(G), χ(G)/ 2 +q} , then there exists a proper k-coloring c of G such that Gc,q is connected, where Gc,q is the subgraph of G such that V(Gc,q ) = V(G) and E(Gc,q ) is the set of edges vw ∈ E(G) that satisfy |c(v)−c(w)| ≥ q.Dado um grafo G = (V(G),E(G)) e um subgrafo H = (V(H),E(H)) de G, uma k-coloração q-backbone de (G,H) é uma função φ : V(G) → { 1 , 2 , 3 ,...,k} tal que, para toda aresta uv ∈ E(G) , temos |φ(u) − φ(v)| ≥ 1 e, para toda aresta uv ∈ E(H) , temos |φ(u) − φ(v)| ≥ q. O número cromático q-backbone de (G,H) , denotado por BBCq (G,H) , é o menor inteiro k tal que existe uma coloração φ como acima. Similarmente, uma k-coloração q-backbone circular de (G,H) é uma função φ : V(G) → { 1 , 2 , 3 ,...,k} tal que, para toda aresta uv ∈ E(G) , temos |φ(u) − φ(v)| ≥ 1 e, para toda aresta uv ∈ E(H) , temos k − q ≥ |φ(u) − φ(v)| ≥ q. O número cromático q-backbone circular de (G,H) , denotado por CBCq (G,H) , é o menor inteiro k tal que existe uma coloração φ como acima. Nesta dissertação, primeiramente apresentamos um breve sumário dos resultados relacionados a Coloração Backbone. Após isto, mostramos que se G é um grafo planar sem ciclos de tamanho quatro e F é uma floresta geradora de caminhos induzidos de G, então CBC2 (G,F) ≤ 7. Por fim, demonstramos o seguinte teorema: se G é um grafo conexo e k ≥ max {χ(G), χ(G)/ 2 + q} , então existe uma k-coloração c de G tal que Gc,q é conexo, onde Gc,q é o subgrafo de G tal que V(Gc,q ) e E(Gc,q ) é formado pelas arestas vw ∈ E(G) que satisfazem |c(v)−c(w)| ≥ q.Circular backbone coloring for graphs without cycles of size fourCircular backbone coloring for graphs without cycles of size fourinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesiscoloração de grafosnúmero cromáticocoloração backbone circulargrafos planares sem C4árvore como backbonegraph coloringchromatic numbercircular backbone coloringplanar graphs without C4tree backboneCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::MATEMATICA APLICADA::MATEMATICA DISCRETA E COMBINATORIAinfo:eu-repo/semantics/openAccessengreponame:Repositório Institucional da Universidade Federal do Ceará (UFC)instname:Universidade Federal do Ceará (UFC)instacron:UFCCezar, Alexandre Azevedohttp://lattes.cnpq.br/5691816023955002https://orcid.org/0000-0001-7074-2753http://lattes.cnpq.br/7659965567201224https://orcid.org/0000-0001-8917-0564http://lattes.cnpq.br/21326146959014162023-12-20ORIGINAL2016_dis_aacezar.pdf2016_dis_aacezar.pdfdissertaçao alexandre azevedoapplication/pdf676173http://repositorio.ufc.br/bitstream/riufc/75456/1/2016_dis_aacezar.pdfff7ccef8fac81d2acb519938319c35b0MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81748http://repositorio.ufc.br/bitstream/riufc/75456/2/license.txt8a4605be74aa9ea9d79846c1fba20a33MD52riufc/754562023-12-21 13:23:48.843oai:repositorio.ufc.br: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Repositório InstitucionalPUBhttp://www.repositorio.ufc.br/ri-oai/requestbu@ufc.br || repositorio@ufc.bropendoar:2023-12-21T16:23:48Repositório Institucional da Universidade Federal do Ceará (UFC) - Universidade Federal do Ceará (UFC)false
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