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- Heawood_conjecture abstract "In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. For surfaces of genus 0, 1, 2, 3, 4, 5, 6, 7, ..., the required number of colors is 4, 7, 8, 9, 10, 11, 12, 12, .... OEIS A000934, the chromatic number or Heawood number.The conjecture was formulated in 1890 by Percy John Heawood and proven in 1968 by Gerhard Ringel and Ted Youngs. One case, the non-orientable Klein bottle, proved an exception to the general formula. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained the conjecture. In other words, Ringel, Youngs and others had to construct extreme examples for every genus g = 1,2,3,.... If g = 12s + k, the genera fall into 12 cases according as k = 0,1,2,3,4,5,6,7,8,9,10,11. To simplify, suppose that case k has been established if only a finite number of g's of the form 12s + k are in doubt. Then the years in which the twelve cases were settled and by whom are the following:1954, Ringel: case 51961, Ringel: cases 3,7,101963, Terry, Welch, Youngs: cases 0,41964, Gustin, Youngs: case 11965, Gustin: case 91966, Youngs: case 61967, Ringel, Youngs: cases 2,8,11The last seven sporadic exceptions were settled as follows:1967, Mayer: cases 18, 20, 231968, Ringel, Youngs: cases 30, 35, 47, 59, and the conjecture was proved.".
- Heawood_conjecture thumbnail Franklin_graph.svg?width=300.
- Heawood_conjecture wikiPageID "1585274".
- Heawood_conjecture wikiPageLength "5479".
- Heawood_conjecture wikiPageOutDegree "35".
- Heawood_conjecture wikiPageRevisionID "670435767".
- Heawood_conjecture wikiPageWikiLink Category:Conjectures.
- Heawood_conjecture wikiPageWikiLink Category:Graph_coloring.
- Heawood_conjecture wikiPageWikiLink Category:Theorems_in_graph_theory.
- Heawood_conjecture wikiPageWikiLink Category:Topological_graph_theory.
- Heawood_conjecture wikiPageWikiLink Complete_graph.
- Heawood_conjecture wikiPageWikiLink Conjecture.
- Heawood_conjecture wikiPageWikiLink Euler_characteristic.
- Heawood_conjecture wikiPageWikiLink Floor_and_ceiling_functions.
- Heawood_conjecture wikiPageWikiLink Four_color_theorem.
- Heawood_conjecture wikiPageWikiLink Franklin_graph.
- Heawood_conjecture wikiPageWikiLink Genus_(mathematics).
- Heawood_conjecture wikiPageWikiLink Gerhard_Ringel.
- Heawood_conjecture wikiPageWikiLink Graph_coloring.
- Heawood_conjecture wikiPageWikiLink Graph_theory.
- Heawood_conjecture wikiPageWikiLink Greedy_coloring.
- Heawood_conjecture wikiPageWikiLink Heawood_graph.
- Heawood_conjecture wikiPageWikiLink John_William_Theodore_Youngs.
- Heawood_conjecture wikiPageWikiLink Kenneth_Appel.
- Heawood_conjecture wikiPageWikiLink Klein_bottle.
- Heawood_conjecture wikiPageWikiLink Orientability.
- Heawood_conjecture wikiPageWikiLink Percy_John_Heawood.
- Heawood_conjecture wikiPageWikiLink Philip_Franklin.
- Heawood_conjecture wikiPageWikiLink Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America.
- Heawood_conjecture wikiPageWikiLink Quarterly_Journal_of_Mathematics.
- Heawood_conjecture wikiPageWikiLink Sphere.
- Heawood_conjecture wikiPageWikiLink Studies_in_Applied_Mathematics.
- Heawood_conjecture wikiPageWikiLink Surface.
- Heawood_conjecture wikiPageWikiLink Torus.
- Heawood_conjecture wikiPageWikiLink Upper_and_lower_bounds.
- Heawood_conjecture wikiPageWikiLink Wikt:necessary.
- Heawood_conjecture wikiPageWikiLink Wolfgang_Haken.
- Heawood_conjecture wikiPageWikiLink File:7x-torus.svg.
- Heawood_conjecture wikiPageWikiLink File:Franklin_graph.svg.
- Heawood_conjecture wikiPageWikiLinkText "Heawood conjecture".
- Heawood_conjecture wikiPageWikiLinkText "Ringel–Youngs theorem".
- Heawood_conjecture title "Heawood Conjecture".
- Heawood_conjecture urlname "HeawoodConjecture".
- Heawood_conjecture wikiPageUsesTemplate Template:Cite_journal.
- Heawood_conjecture wikiPageUsesTemplate Template:Hdl.
- Heawood_conjecture wikiPageUsesTemplate Template:Mathworld.
- Heawood_conjecture wikiPageUsesTemplate Template:OEIS2C.
- Heawood_conjecture subject Category:Conjectures.
- Heawood_conjecture subject Category:Graph_coloring.
- Heawood_conjecture subject Category:Theorems_in_graph_theory.
- Heawood_conjecture subject Category:Topological_graph_theory.
- Heawood_conjecture type Conjecture.
- Heawood_conjecture type Redirect.
- Heawood_conjecture type Statement.
- Heawood_conjecture type Theorem.
- Heawood_conjecture type Statement.
- Heawood_conjecture comment "In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. For surfaces of genus 0, 1, 2, 3, 4, 5, 6, 7, ..., the required number of colors is 4, 7, 8, 9, 10, 11, 12, 12, .... OEIS A000934, the chromatic number or Heawood number.The conjecture was formulated in 1890 by Percy John Heawood and proven in 1968 by Gerhard Ringel and Ted Youngs.".
- Heawood_conjecture label "Heawood conjecture".
- Heawood_conjecture sameAs Q2799491.
- Heawood_conjecture sameAs Conjectura_de_Heawood.
- Heawood_conjecture sameAs Satz_von_Ringel-Youngs.
- Heawood_conjecture sameAs m.05dflx.
- Heawood_conjecture sameAs Q2799491.
- Heawood_conjecture wasDerivedFrom Heawood_conjecture?oldid=670435767.
- Heawood_conjecture depiction Franklin_graph.svg.
- Heawood_conjecture isPrimaryTopicOf Heawood_conjecture.