A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Tipo de documento: | preprint |
Idioma: | eng |
Título da fonte: | SciELO Preprints |
Texto Completo: | https://preprints.scielo.org/index.php/scielo/preprint/view/7216 |
Resumo: | The Four-Color Theorem originated from the attempt to solve the problem of painting MAPS over a plane or spherical surface. Over a century and a half, this problem underwent various abstractions until it was resolved in 1976. The proposed solution, which is disruptive, computationally calculates the number of possible states for a representation of a flat map. Although it is resolved, the lack of a formal proof for this problem causes some discomfort. Therefore, a solution that uses more traditional techniques and is easily understandable is needed. In a previous article, a solution based on equalities and inequalities between boundaries was presented. Now in this article, a generic spheroidal MAP is subjected to various one-to-one relationships until a generator of all possible MAPS on a two-dimensional surface partitioned into n2 cells are found. Four-Colors are proved to be necessary and sufficient to paint a two dimensional MAP. It is explained at the end of the article that the imposition of a fifth color as a necessary condition implies a contradiction. |
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A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREMFour-Color Theoremdiscrete method solutionDiscriminant functionThe Four-Color Theorem originated from the attempt to solve the problem of painting MAPS over a plane or spherical surface. Over a century and a half, this problem underwent various abstractions until it was resolved in 1976. The proposed solution, which is disruptive, computationally calculates the number of possible states for a representation of a flat map. Although it is resolved, the lack of a formal proof for this problem causes some discomfort. Therefore, a solution that uses more traditional techniques and is easily understandable is needed. In a previous article, a solution based on equalities and inequalities between boundaries was presented. Now in this article, a generic spheroidal MAP is subjected to various one-to-one relationships until a generator of all possible MAPS on a two-dimensional surface partitioned into n2 cells are found. Four-Colors are proved to be necessary and sufficient to paint a two dimensional MAP. It is explained at the end of the article that the imposition of a fifth color as a necessary condition implies a contradiction.SciELO PreprintsSciELO PreprintsSciELO Preprints2023-11-07info:eu-repo/semantics/preprintinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://preprints.scielo.org/index.php/scielo/preprint/view/721610.1590/SciELOPreprints.7216enghttps://preprints.scielo.org/index.php/scielo/article/view/7216/13676Copyright (c) 2023 José Ulisses Jansenhttps://creativecommons.org/licenses/by/4.0info:eu-repo/semantics/openAccessJansen, José Ulissesreponame:SciELO Preprintsinstname:Scientific Electronic Library Online (SCIELO)instacron:SCI2023-10-31T11:05:29Zoai:ops.preprints.scielo.org:preprint/7216Servidor de preprintshttps://preprints.scielo.org/index.php/scieloONGhttps://preprints.scielo.org/index.php/scielo/oaiscielo.submission@scielo.orgopendoar:2023-10-31T11:05:29SciELO Preprints - Scientific Electronic Library Online (SCIELO)false |
dc.title.none.fl_str_mv |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
title |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
spellingShingle |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM Jansen, José Ulisses Four-Color Theorem discrete method solution Discriminant function |
title_short |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
title_full |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
title_fullStr |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
title_full_unstemmed |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
title_sort |
A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM |
author |
Jansen, José Ulisses |
author_facet |
Jansen, José Ulisses |
author_role |
author |
dc.contributor.author.fl_str_mv |
Jansen, José Ulisses |
dc.subject.por.fl_str_mv |
Four-Color Theorem discrete method solution Discriminant function |
topic |
Four-Color Theorem discrete method solution Discriminant function |
description |
The Four-Color Theorem originated from the attempt to solve the problem of painting MAPS over a plane or spherical surface. Over a century and a half, this problem underwent various abstractions until it was resolved in 1976. The proposed solution, which is disruptive, computationally calculates the number of possible states for a representation of a flat map. Although it is resolved, the lack of a formal proof for this problem causes some discomfort. Therefore, a solution that uses more traditional techniques and is easily understandable is needed. In a previous article, a solution based on equalities and inequalities between boundaries was presented. Now in this article, a generic spheroidal MAP is subjected to various one-to-one relationships until a generator of all possible MAPS on a two-dimensional surface partitioned into n2 cells are found. Four-Colors are proved to be necessary and sufficient to paint a two dimensional MAP. It is explained at the end of the article that the imposition of a fifth color as a necessary condition implies a contradiction. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023-11-07 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/preprint info:eu-repo/semantics/publishedVersion |
format |
preprint |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://preprints.scielo.org/index.php/scielo/preprint/view/7216 10.1590/SciELOPreprints.7216 |
url |
https://preprints.scielo.org/index.php/scielo/preprint/view/7216 |
identifier_str_mv |
10.1590/SciELOPreprints.7216 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://preprints.scielo.org/index.php/scielo/article/view/7216/13676 |
dc.rights.driver.fl_str_mv |
Copyright (c) 2023 José Ulisses Jansen https://creativecommons.org/licenses/by/4.0 info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Copyright (c) 2023 José Ulisses Jansen https://creativecommons.org/licenses/by/4.0 |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
SciELO Preprints SciELO Preprints SciELO Preprints |
publisher.none.fl_str_mv |
SciELO Preprints SciELO Preprints SciELO Preprints |
dc.source.none.fl_str_mv |
reponame:SciELO Preprints instname:Scientific Electronic Library Online (SCIELO) instacron:SCI |
instname_str |
Scientific Electronic Library Online (SCIELO) |
instacron_str |
SCI |
institution |
SCI |
reponame_str |
SciELO Preprints |
collection |
SciELO Preprints |
repository.name.fl_str_mv |
SciELO Preprints - Scientific Electronic Library Online (SCIELO) |
repository.mail.fl_str_mv |
scielo.submission@scielo.org |
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1797047813724438528 |