Sobre b-coloraÃÃo de grafos com cintura pelo menos 6
Autor(a) principal: | |
---|---|
Data de Publicação: | 2013 |
Tipo de documento: | Dissertação |
Idioma: | por |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da UFC |
Texto Completo: | http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=9630 |
Resumo: | The coloring problem is among the most studied in the Graph Theory due to its great theoretical and practical importance. Since the problem of coloring the vertices of a graph G either with the smallest amount of colors is NP-hard, various coloring heuristics are examined to obtain a proper colouring with a reasonably small number of colors. Given a graph G, the b heuristic of colouring comes down to decrease the amount of colors in a proper colouring c, so that, if all vertices of a color class fail to see any color in your neighborhood, then we can change the color to any color these vertices nonexistent in your neighborhood. Thus, we obtain a coloring c ′ with a color unless c. Irving and Molove defined the b-coloring of a graph G as a coloring where every color class has a vertex that is adjacent the other color classes. These vertices are called b-vertices. Irving and Molove also defined the b-chromatic number as the largest integer k, such that G admits a b-coloring by k colors. They showed that determine the value of the b-chromatic number of any graph is NP-hard, but polynomial for trees. Irving and Molove also defined the m-degree of a graph, which is the largest integer m(G) such that there are m(G) vertices with degree at least m(G) − 1. Irving and Molove showed that the m-degree is an upper limit to the b-chromatic number and showed that it is m(T) or m(T)−1 to every tree T, where its value is m(T) if, and only if, T has a good set. In this dissertation, we analyze the relationship between the girth, which is the size of the smallest cycle, and the b-chromatic number of a graph G. More specifically, we try to find the smallest integer g ∗ such that if the girth of G is at least g ∗ , then the b-chromatic number equals m(G) or m(G)−1. Show that the value of g ∗ is at most 6 could be an important step in demonstrating the famous conjecture of Erd˝os-Faber-LovÂasz, but the best known upper limit to g ∗ is 9. We characterize the graphs whose girth is at least 6 and not have a good set and show how b-color them optimally. Furthermore, we show how b-color, also optimally, graphs whose girth is at least 7 and not have good set. |
id |
UFC_4f397431b54b7f629e6227f69ab6e4e5 |
---|---|
oai_identifier_str |
oai:www.teses.ufc.br:6404 |
network_acronym_str |
UFC |
network_name_str |
Biblioteca Digital de Teses e Dissertações da UFC |
spelling |
info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisSobre b-coloraÃÃo de grafos com cintura pelo menos 6About b-coloring of graphs with waist at least 62013-02-25Victor Almeida Campos86698974315http://lattes.cnpq.br/0802023762311924Ana Shirley Ferreira da Silva62219340368http://lattes.cnpq.br/2132614695901416Rudini Menezes Sampaio25240703876http://lattes.cnpq.br/2845950448235863 Erika Morais Martins Coelho96687975153 http://lattes.cnpq.br/938948701593850901458225321http://lattes.cnpq.br/9245122165772401Carlos Vinicius Gomes Costa LimaUniversidade Federal do CearÃPrograma de PÃs-GraduaÃÃo em CiÃncia da ComputaÃÃoUFCBRb-coloraÃÃo Cintura Conjectura de Erdos-Faber-LovÃsz Conjunto bomb-Colouring, Girth, Good set, Conjecture of ErdÃs-Faber-LovÃszCIENCIA DA COMPUTACAOThe coloring problem is among the most studied in the Graph Theory due to its great theoretical and practical importance. Since the problem of coloring the vertices of a graph G either with the smallest amount of colors is NP-hard, various coloring heuristics are examined to obtain a proper colouring with a reasonably small number of colors. Given a graph G, the b heuristic of colouring comes down to decrease the amount of colors in a proper colouring c, so that, if all vertices of a color class fail to see any color in your neighborhood, then we can change the color to any color these vertices nonexistent in your neighborhood. Thus, we obtain a coloring c ′ with a color unless c. Irving and Molove defined the b-coloring of a graph G as a coloring where every color class has a vertex that is adjacent the other color classes. These vertices are called b-vertices. Irving and Molove also defined the b-chromatic number as the largest integer k, such that G admits a b-coloring by k colors. They showed that determine the value of the b-chromatic number of any graph is NP-hard, but polynomial for trees. Irving and Molove also defined the m-degree of a graph, which is the largest integer m(G) such that there are m(G) vertices with degree at least m(G) − 1. Irving and Molove showed that the m-degree is an upper limit to the b-chromatic number and showed that it is m(T) or m(T)−1 to every tree T, where its value is m(T) if, and only if, T has a good set. In this dissertation, we analyze the relationship between the girth, which is the size of the smallest cycle, and the b-chromatic number of a graph G. More specifically, we try to find the smallest integer g ∗ such that if the girth of G is at least g ∗ , then the b-chromatic number equals m(G) or m(G)−1. Show that the value of g ∗ is at most 6 could be an important step in demonstrating the famous conjecture of Erd˝os-Faber-LovÂasz, but the best known upper limit to g ∗ is 9. We characterize the graphs whose girth is at least 6 and not have a good set and show how b-color them optimally. Furthermore, we show how b-color, also optimally, graphs whose girth is at least 7 and not have good set.O problema de coloraÃÃo està entre os mais estudados dentro da Teoria dos Grafos devido a sua grande importÃncia teorica e prÃtica. Dado que o problema de colorir os vÃrtices de um grafo G qualquer com a menor quantidade de cores à NP-difÃcil, vÃrias heurÃsticas de coloraÃÃo sÃo estudadas a fim de obter uma coloraÃÃo prÃpria com um nÃmero de cores razoavelmente pequeno. Dado um grafo G, a heurÃstica b de coloraÃÃo se resume a diminuir a quantidade de cores utilizadas em uma coloraÃÃo prÃpria c, de modo que, se todos os vÃrtices de uma classe de cor deixam de ver alguma cor em sua vizinhanÃa, entÃo podemos modificar a cor desses vÃrtices para qualquer cor inexistente em sua vizinhanÃa. Dessa forma, obtemos uma coloraÃÃo c′ com uma cor a menos que c. Irving e Molove definiram a b-coloraÃÃo de um grafo G como uma coloraÃÃo onde toda classe de cor possui um vÃrtice que à adjacente as demais classes de cor. Esses vÃrtices sÃo chamados b-vÃrtices. Irving e Molove tambÃm definiram o nÃmero b-cromÃtico como o maior inteiro k tal que G admite uma b-coloraÃÃo por k cores. Eles mostraram que determinar o nÃmero b-cromÃtico de um grafo qualquer à um problema NP-difÃcil, mas polinomial para Ãrvores. Irving e Molove tambÃm definiram o m-grau de um grafo, que à o maior inteiro m(G) tal que existem m(G) vÃrtices com grau pelo menos m(G)−1. Irving e Molove mostraram que o m-grau à um limite superior para nÃmero b-cromÃtico e mostraram que o mesmo à igual a m(T) ou a m(T)−1, para toda Ãrvore T, onde o nÃmero b-cromÃtico à igual a m(T) se, e somente se, T possui um conjunto bom. Nesta dissertaÃÃo, verificamos a relaÃÃo entre a cintura, que à o tamanho do menor ciclo, e o nÃmero b-cromÃtico de um grafo G. Mais especificamente, tentamos encontrar o menor inteiro g∗ tal que, se a cintura de G à pelo menos g∗, entÃo o nÃmero b-cromÃtico à igual a m(G) ou m(G)−1. Mostrar que o valor de g∗ à no mÃximo 6 poderia ser um passo importante para demonstrar a famosa Conjectura de ErdÃs-Faber-Lovasz, mas o melhor limite superior conhecido para g∗ à 9. Caracterizamos os grafos cuja cintura à pelo menos 6 e nÃo possuem um conjunto bom e mostramos como b-colori-los de forma Ãtima. AlÃm disso, mostramos como bicolorir, tambÃm de forma Ãtima, os grafos cuja cintura à pelo menos 7 e nÃo possuem conjunto bom.Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgicohttp://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=9630application/pdfinfo:eu-repo/semantics/openAccessporreponame:Biblioteca Digital de Teses e Dissertações da UFCinstname:Universidade Federal do Cearáinstacron:UFC2019-01-21T11:22:54Zmail@mail.com - |
dc.title.pt.fl_str_mv |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
dc.title.alternative.en.fl_str_mv |
About b-coloring of graphs with waist at least 6 |
title |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
spellingShingle |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 Carlos Vinicius Gomes Costa Lima b-coloraÃÃo Cintura Conjectura de Erdos-Faber-LovÃsz Conjunto bom b-Colouring, Girth, Good set, Conjecture of ErdÃs-Faber-LovÃsz CIENCIA DA COMPUTACAO |
title_short |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
title_full |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
title_fullStr |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
title_full_unstemmed |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
title_sort |
Sobre b-coloraÃÃo de grafos com cintura pelo menos 6 |
author |
Carlos Vinicius Gomes Costa Lima |
author_facet |
Carlos Vinicius Gomes Costa Lima |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Victor Almeida Campos |
dc.contributor.advisor1ID.fl_str_mv |
86698974315 |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/0802023762311924 |
dc.contributor.referee1.fl_str_mv |
Ana Shirley Ferreira da Silva |
dc.contributor.referee1ID.fl_str_mv |
62219340368 |
dc.contributor.referee1Lattes.fl_str_mv |
http://lattes.cnpq.br/2132614695901416 |
dc.contributor.referee2.fl_str_mv |
Rudini Menezes Sampaio |
dc.contributor.referee2ID.fl_str_mv |
25240703876 |
dc.contributor.referee2Lattes.fl_str_mv |
http://lattes.cnpq.br/2845950448235863 |
dc.contributor.referee3.fl_str_mv |
Erika Morais Martins Coelho |
dc.contributor.referee3ID.fl_str_mv |
96687975153 |
dc.contributor.referee3Lattes.fl_str_mv |
http://lattes.cnpq.br/9389487015938509 |
dc.contributor.authorID.fl_str_mv |
01458225321 |
dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/9245122165772401 |
dc.contributor.author.fl_str_mv |
Carlos Vinicius Gomes Costa Lima |
contributor_str_mv |
Victor Almeida Campos Ana Shirley Ferreira da Silva Rudini Menezes Sampaio Erika Morais Martins Coelho |
dc.subject.por.fl_str_mv |
b-coloraÃÃo Cintura Conjectura de Erdos-Faber-LovÃsz Conjunto bom |
topic |
b-coloraÃÃo Cintura Conjectura de Erdos-Faber-LovÃsz Conjunto bom b-Colouring, Girth, Good set, Conjecture of ErdÃs-Faber-LovÃsz CIENCIA DA COMPUTACAO |
dc.subject.eng.fl_str_mv |
b-Colouring, Girth, Good set, Conjecture of ErdÃs-Faber-LovÃsz |
dc.subject.cnpq.fl_str_mv |
CIENCIA DA COMPUTACAO |
dc.description.sponsorship.fl_txt_mv |
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico |
dc.description.abstract.por.fl_txt_mv |
The coloring problem is among the most studied in the Graph Theory due to its great theoretical and practical importance. Since the problem of coloring the vertices of a graph G either with the smallest amount of colors is NP-hard, various coloring heuristics are examined to obtain a proper colouring with a reasonably small number of colors. Given a graph G, the b heuristic of colouring comes down to decrease the amount of colors in a proper colouring c, so that, if all vertices of a color class fail to see any color in your neighborhood, then we can change the color to any color these vertices nonexistent in your neighborhood. Thus, we obtain a coloring c ′ with a color unless c. Irving and Molove defined the b-coloring of a graph G as a coloring where every color class has a vertex that is adjacent the other color classes. These vertices are called b-vertices. Irving and Molove also defined the b-chromatic number as the largest integer k, such that G admits a b-coloring by k colors. They showed that determine the value of the b-chromatic number of any graph is NP-hard, but polynomial for trees. Irving and Molove also defined the m-degree of a graph, which is the largest integer m(G) such that there are m(G) vertices with degree at least m(G) − 1. Irving and Molove showed that the m-degree is an upper limit to the b-chromatic number and showed that it is m(T) or m(T)−1 to every tree T, where its value is m(T) if, and only if, T has a good set. In this dissertation, we analyze the relationship between the girth, which is the size of the smallest cycle, and the b-chromatic number of a graph G. More specifically, we try to find the smallest integer g ∗ such that if the girth of G is at least g ∗ , then the b-chromatic number equals m(G) or m(G)−1. Show that the value of g ∗ is at most 6 could be an important step in demonstrating the famous conjecture of Erd˝os-Faber-LovÂasz, but the best known upper limit to g ∗ is 9. We characterize the graphs whose girth is at least 6 and not have a good set and show how b-color them optimally. Furthermore, we show how b-color, also optimally, graphs whose girth is at least 7 and not have good set. O problema de coloraÃÃo està entre os mais estudados dentro da Teoria dos Grafos devido a sua grande importÃncia teorica e prÃtica. Dado que o problema de colorir os vÃrtices de um grafo G qualquer com a menor quantidade de cores à NP-difÃcil, vÃrias heurÃsticas de coloraÃÃo sÃo estudadas a fim de obter uma coloraÃÃo prÃpria com um nÃmero de cores razoavelmente pequeno. Dado um grafo G, a heurÃstica b de coloraÃÃo se resume a diminuir a quantidade de cores utilizadas em uma coloraÃÃo prÃpria c, de modo que, se todos os vÃrtices de uma classe de cor deixam de ver alguma cor em sua vizinhanÃa, entÃo podemos modificar a cor desses vÃrtices para qualquer cor inexistente em sua vizinhanÃa. Dessa forma, obtemos uma coloraÃÃo c′ com uma cor a menos que c. Irving e Molove definiram a b-coloraÃÃo de um grafo G como uma coloraÃÃo onde toda classe de cor possui um vÃrtice que à adjacente as demais classes de cor. Esses vÃrtices sÃo chamados b-vÃrtices. Irving e Molove tambÃm definiram o nÃmero b-cromÃtico como o maior inteiro k tal que G admite uma b-coloraÃÃo por k cores. Eles mostraram que determinar o nÃmero b-cromÃtico de um grafo qualquer à um problema NP-difÃcil, mas polinomial para Ãrvores. Irving e Molove tambÃm definiram o m-grau de um grafo, que à o maior inteiro m(G) tal que existem m(G) vÃrtices com grau pelo menos m(G)−1. Irving e Molove mostraram que o m-grau à um limite superior para nÃmero b-cromÃtico e mostraram que o mesmo à igual a m(T) ou a m(T)−1, para toda Ãrvore T, onde o nÃmero b-cromÃtico à igual a m(T) se, e somente se, T possui um conjunto bom. Nesta dissertaÃÃo, verificamos a relaÃÃo entre a cintura, que à o tamanho do menor ciclo, e o nÃmero b-cromÃtico de um grafo G. Mais especificamente, tentamos encontrar o menor inteiro g∗ tal que, se a cintura de G à pelo menos g∗, entÃo o nÃmero b-cromÃtico à igual a m(G) ou m(G)−1. Mostrar que o valor de g∗ à no mÃximo 6 poderia ser um passo importante para demonstrar a famosa Conjectura de ErdÃs-Faber-Lovasz, mas o melhor limite superior conhecido para g∗ à 9. Caracterizamos os grafos cuja cintura à pelo menos 6 e nÃo possuem um conjunto bom e mostramos como b-colori-los de forma Ãtima. AlÃm disso, mostramos como bicolorir, tambÃm de forma Ãtima, os grafos cuja cintura à pelo menos 7 e nÃo possuem conjunto bom. |
description |
The coloring problem is among the most studied in the Graph Theory due to its great theoretical and practical importance. Since the problem of coloring the vertices of a graph G either with the smallest amount of colors is NP-hard, various coloring heuristics are examined to obtain a proper colouring with a reasonably small number of colors. Given a graph G, the b heuristic of colouring comes down to decrease the amount of colors in a proper colouring c, so that, if all vertices of a color class fail to see any color in your neighborhood, then we can change the color to any color these vertices nonexistent in your neighborhood. Thus, we obtain a coloring c ′ with a color unless c. Irving and Molove defined the b-coloring of a graph G as a coloring where every color class has a vertex that is adjacent the other color classes. These vertices are called b-vertices. Irving and Molove also defined the b-chromatic number as the largest integer k, such that G admits a b-coloring by k colors. They showed that determine the value of the b-chromatic number of any graph is NP-hard, but polynomial for trees. Irving and Molove also defined the m-degree of a graph, which is the largest integer m(G) such that there are m(G) vertices with degree at least m(G) − 1. Irving and Molove showed that the m-degree is an upper limit to the b-chromatic number and showed that it is m(T) or m(T)−1 to every tree T, where its value is m(T) if, and only if, T has a good set. In this dissertation, we analyze the relationship between the girth, which is the size of the smallest cycle, and the b-chromatic number of a graph G. More specifically, we try to find the smallest integer g ∗ such that if the girth of G is at least g ∗ , then the b-chromatic number equals m(G) or m(G)−1. Show that the value of g ∗ is at most 6 could be an important step in demonstrating the famous conjecture of Erd˝os-Faber-LovÂasz, but the best known upper limit to g ∗ is 9. We characterize the graphs whose girth is at least 6 and not have a good set and show how b-color them optimally. Furthermore, we show how b-color, also optimally, graphs whose girth is at least 7 and not have good set. |
publishDate |
2013 |
dc.date.issued.fl_str_mv |
2013-02-25 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
status_str |
publishedVersion |
format |
masterThesis |
dc.identifier.uri.fl_str_mv |
http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=9630 |
url |
http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=9630 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Universidade Federal do Cearà |
dc.publisher.program.fl_str_mv |
Programa de PÃs-GraduaÃÃo em CiÃncia da ComputaÃÃo |
dc.publisher.initials.fl_str_mv |
UFC |
dc.publisher.country.fl_str_mv |
BR |
publisher.none.fl_str_mv |
Universidade Federal do Cearà |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da UFC instname:Universidade Federal do Ceará instacron:UFC |
reponame_str |
Biblioteca Digital de Teses e Dissertações da UFC |
collection |
Biblioteca Digital de Teses e Dissertações da UFC |
instname_str |
Universidade Federal do Ceará |
instacron_str |
UFC |
institution |
UFC |
repository.name.fl_str_mv |
-
|
repository.mail.fl_str_mv |
mail@mail.com |
_version_ |
1643295173685280768 |