Meta tags:

Headings (most frequently used words):

of, proof, mathematics, four, color, theorem, contents, formulation, history, summary, ideas, false, disproofs, three, coloring, generalizations, relation, to, other, areas, use, outside, see, also, notes, references, external, links, early, attempts, by, computer, simplification, and, verification, infinite, graphs, higher, surfaces, solid, regions,

Text of the page (most frequently used words):
the (405), and (116), four (96), that (75), color (63), #theorem (61), colors (58), with (57), #regions (55), map (53), proof (53), this (45), graph (41), for (40), are (39), can (38), not (36), planar (33), was (32), one (31), vertex (27), colored (27), coloring (26), doi (25), from (24), vertices (24), haken (23), number (23), edit (22), appel (22), set (22), configuration (22), graphs (21), mathematics (21), two (21), more (21), every (20), only (20), three (19), same (19), other (19), which (19), wilson (18), problem (18), such (18), degree (18), kempe (17), adjacent (17), each (17), computer (16), all (16), then (16), configurations (16), 2014 (15), but (15), have (15), also (15), has (15), thomas (14), any (14), region (14), theory (13), mathematical (13), maps (13), colorable (13), were (13), original (12), there (12), neighbors (12), reducible (12), may (11), using (11), isbn (11), vol (11), some (11), plane (11), since (11), they (11), red (11), robin (10), pdf (10), new (10), history (10), conjecture (10), false (10), cannot (10), without (10), first (10), five (10), its (10), unavoidable (10), american (9), case (9), requires (9), blue (9), must (9), require (9), because (9), states (9), would (9), boundary (9), edges (9), displaystyle (9), where (9), above (9), colour (8), university (8), surfaces (8), heawood (8), discharging (8), morgan (8), large (8), ring (8), wikipedia (7), terms (7), non (7), use (7), different (7), society (7), press (7), 978 (7), infinite (7), kenneth (7), called (7), their (7), common (7), least (7), needed (7), given (7), used (7), six (7), torus (7), formula (7), together (7), charge (7), proving (7), minimal (7), page (6), john (6), tait (6), 1998 (6), archived (6), robertson (6), ringel (6), journal (6), gonthier (6), checked (6), 1989 (6), wolfgang (6), simple (6), country (6), france (6), thus (6), touches (6), guthrie (6), surface (6), most (6), seven (6), into (6), based (6), genus (6), than (6), example (6), prove (6), single (6), fact (6), counterexample (6), initially (6), argument (6), time (6), had (6), search (5), statement (5), about (5), proofs (5), princeton (5), 1880 (5), jstor (5), 2307 (5), york (5), 1997 (5), 1996 (5), 2008 (5), 1890 (5), hadwiger (5), reducibility (5), illinois (5), complete (5), way (5), many (5), article (5), contiguous (5), known (5), even (5), share (5), necessary (5), move (5), euler (5), required (5), proved (5), shown (5), positive (5), finite (5), differently (5), however (5), these (5), remaining (5), possible (5), over (5), procedure (5), could (5), early (5), his (5), found (5), toggle (4), contents (4), apply (4), assisted (4), generalizations (4), notices (4), 1852 (4), links (4), eds (4), note (4), tietze (4), paul (4), seymour (4), sanders (4), springer (4), 2011 (4), 2003 (4), martin (4), 1879 (4), georges (4), 2005 (4), how (4), part (4), 1977 (4), another (4), corner (4), points (4), netherlands (4), those (4), when (4), divided (4), areas (4), drawn (4), faces (4), requiring (4), characteristic (4), both (4), several (4), joining (4), here (4), while (4), disproofs (4), arbitrarily (4), work (4), long (4), like (4), published (4), still (4), sum (4), general (4), chain (4), smaller (4), triangulated (4), tools (4), later (4), unavoidability (4), portion (4), francis (4), hide (4), sidebar (4), add (3), view (3), contact (3), policy (3), short (3), description (3), retrieved (3), 1090 (3), 1976 (3), 2001 (3), oxford (3), 2002 (3), suffice (3), 1999 (3), 222 (3), 1980 (3), 1986 (3), daniel (3), neil (3), s2cid (3), proceedings (3), computing (3), solution (3), bibcode (3), book (3), 1016 (3), williams (3), 1943 (3), gethner (3), 1854 (3), athenaeum (3), 1007 (3), publishing (3), borodin (3), 1984 (3), 108 (3), bar (3), natan (3), koch (3), island (3), open (3), colorability (3), discovery (3), trust (3), notion (3), see (3), math (3), möbius (3), whose (3), connected (3), point (3), free (3), total (3), particular (3), according (3), does (3), world (3), united (3), equivalent (3), result (3), solid (3), rod (3), frederick (3), unbounded (3), distinct (3), projective (3), form (3), strip (3), klein (3), bottle (3), denote (3), double (3), gives (3), edge (3), proposed (3), orientable (3), consider (3), maximum (3), chi (3), crossings (3), order (3), touching (3), them (3), needs (3), complexity (3), generalized (3), though (3), valid (3), create (3), before (3), 400 (3), rules (3), remains (3), triangular (3), good (3), assistance (3), removed (3), showed (3), alternating (3), now (3), following (3), summary (3), although (3), formulation (3), ideas (3), need (3), algorithm (3), hand (3), rumors (3), schmidt (3), pages (3), mathematicians (3), who (3), reduced (3), took (3), attempts (3), segment (3), receive (3), main (3), languages (2), table (2), privacy (2), under (2), wikimedia (2), inc (2), commons (2), 2026 (2), categories (2), wikidata (2), theorems (2), 216 (2), 1994 (2), encyclopedia (2), external (2), david (2), 691 (2), america (2), remarks (2), colourings (2), 1017 (2), proc (2), soc (2), recent (2), london (2), cambridge (2), 1910 (2), 159 (2), der (2), über (2), heinrich (2), 1995 (2), 848 (2), swart (2), 2321855 (2), monthly (2), 1968 (2), 438 (2), youngs (2), 1974 (2), allwright (2), 2013 (2), reed (2), review (2), pegg (2), 1967 (2), nash (2), 2012 (2), arxiv (2), mckay (2), magnant (2), space (2), 161 (2), geographical (2), colours (2), 2369235 (2), hudson (2), 3647828 (2), 133 (2), hugo (2), unpublished (2), 2009 (2), involve (2), 249 (2), ellen (2), congr (2), numer (2), june (2), fritsch (2), german (2), idea (2), 1799998 (2), cayley (2), arthur (2), face (2), frank (2), lie (2), algebras (2), dror (2), october (2), 1215 (2), ijm (2), references (2), 1572306 (2), coxeter (2), moon (2), books (2), 319 (2), state (2), discrete (2), regular (2), 852 (2), 853 (2), 1860 (2), critical (2), augustus (2), donald (2), 1897 (2), respective (2), closures (2), notes (2), network (2), portal (2), itself (2), germany (2), belgium (2), assigned (2), border (2), property (2), rest (2), saint (2), alaska (2), countries (2), between (2), outside (2), relation (2), dimensional (2), examples (2), cuboids (2), considered (2), words (2), represented (2), crossing (2), curves (2), chromatic (2), gerhard (2), szilassi (2), polyhedron (2), real (2), giving (2), petersen (2), subdivision (2), mutually (2), around (2), along (2), lines (2), pair (2), again (2), certain (2), top (2), after (2), implies (2), left (2), lfloor (2), frac (2), sqrt (2), right (2), rfloor (2), follows (2), higher (2), applies (2), generally (2), subgraph (2), stating (2), arrows (2), neighboring (2), landlocked (2), arbitrary (2), violate (2), parts (2), perhaps (2), underlying (2), restriction (2), appear (2), counterexamples (2), always (2), drawing (2), second (2), stood (2), never (2), technical (2), modified (2), introducing (2), extremely (2), complex (2), volume (2), done (2), years (2), systematically (2), restrict (2), primary (2), method (2), intuitive (2), electrical (2), cases (2), colorings (2), extended (2), fewer (2), turn (2), removing (2), considering (2), situation (2), consisting (2), having (2), suffices (2), added (2), well (2), occur (2), show (2), flawed (2), noticed (2), through (2), remove (2), green (2), yellow (2), path (2), paths (2), intersect (2), suppose (2), back (2), less (2), shared (2), boundaries (2), make (2), including (2), modern (2), programs (2), coq (2), approach (2), efficient (2), 633 (2), check (2), announced (2), snark (2), simplification (2), verification (2), flaw (2), error (2), asked (2), detailed (2), microfiche (2), others (2), widely (2), department (2), major (2), human (2), accepted (2), demonstration (2), did (2), out (2), stewart (2), very (2), exist (2), computers (2), verified (2), methods (2), developed (2), aided (2), far (2), terminology (2), until (2), incorrect (2), unless (2), counties (2), inclosed (2), believe (2), upon (2), elementary (2), question (2), student (2), wanted (2), line (2), simpler (2), theoretic (2), allowed (2), version (2), proven (2), appearance (2), upload (2), file (2), changes (2), read (2), english (2), subsection (2), log (2), account (2), donate (2), menu (2), topic, mobile, cookie, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, text, available, additional, site, you, agree, registered, trademark, profit, organization, foundation, creative, attribution, sharealike, license, rendered, parsoid, last, edited, september, utc, hidden, articles, statements, https, org, index, php, title, four_color_theorem, oldid, 1375914031, mathoverflow, list, march, 228, noti3305, ems, watkins, parks, 2023, 19402, science, library, jersey, 3235839, 15822, 729, s0370164600044643, edinburgh, excluded, minor, lamb, preece, lecture, series, 267, 1725004, 521, 65376, cbo9780511721335, 201, surveys, combinatorics, sided, 155, jahresbericht, deutschen, mathematiker, vereinigung, einige, bemerkungen, das, des, kartenfärbens, auf, einseitigen, flächen, 859, 2000, 1633714, update, edward, reinier, association, 702, 0602826, 697, philosophical, implications, dover, publications, 486, 65092, assaults, conquest, kainen, saaty, 1441258, 1006, jctb, 1750, combin, ser, efficiently, 575, 14962541, 1427555, 89791, 785, 1145, 237814, 238005, 571, 28th, acm, symposium, stoc, 445, 16591648, pmid, 225066, pmc, 1073, pnas, 1968pnas, 438r, natl, acad, sci, usa, berlin, verlag, industry, studies, painting, office, bruce, melendez, berenguer, sendra, hernandez, del, pino, 1086, 1084, colossal, connor, mactutor, archive, survey, 301, 0214501, s0021, 9800, 80077, 286, combinatorial, 2012arxiv1201, 2852m, 1201, 2852, brendan, rectangular, blocks, 170, 7151, dmgt, 1535, discussiones, mathematicae, 220, 193, hud, 423, 417, 110, 338, 332, quarterly, pure, applied, eine, klassifikation, streckenkomplexe, 143, vierteljschr, naturforsch, ges, zürich, 1393, 2463991, 1382, formal, 2017, kalichanda, bopanna, mentis, alexander, 265, 2140, 175, 1050, 05049, zbl, 2050581, 164, 726, tinting, rudolf, gerda, translated, julie, peschke, 1633950, 387, 98497, 4612, 1720, topological, foundations, blackwell, 261, 259, royal, 0832128, metody, diskretnogo, analiza, bernhart, digest, 225, 0465921, 1002, jgt, 3190010305, 207, 2103049, 1466574, bf01196130, alg, 9606016, combinatorica, contemporary, collaboration, providence, rhode, 8735627, 1025335, 8218, 5103, conm, 098, 237, 121, 1038, scientificamerican1077, 1977sciam, 237d, 108a, scientific, 567, 0543793, 1256049012, 491, 490, 0543792, 1256049011, 429, allaire, 1978, mccarthy, winnipeg, man, utilitas, mathematica, 0535003, 919628, 7th, manitoba, conference, numerical, summer, 1971, 277, 273, leonardo, 2018, beyond, hedetniemi, stephen, international, 3930641, 97684, 97686, 0_11, 115, favorite, conjectures, problems, haynes, teresa, gera, ralucca, steinberg, richard, 1993, gimbel, kennedy, quintas, louis, annals, amsterdam, north, holland, 248, 1217995, 444, 89441, s0167, 5060, 70391, 211, quo, vadis, dailey, uniqueness, 293, 0012, 365x, 90236, 289, 165, 162, 157, 150, 153, 105, 107, 145, 146, gary, chartrand, linda, lesniak, crc, 221, digraphs, 139, 142, 1960, recreations, essays, macmillan, 232, rouse, ball, april, philosophy, chapters, historical, whewell, 503, 501, anonymous, mackenzie, mit, 2004, p103, mechanizing, risk, folk, lore, originated, seems, erroneous, 257, s0002, 9904, 00421, bull, amer, maddison, isabel, lloyd, keith, 853916, 116, 1736, 1936, biggs, norman, 849, definitions, pairwise, disjoint, subsets, sets, belongs, unit, distance, apart, nelson, triangle, grötzsch, apollonian, fails, bodies, water, europe, luxembourg, fifth, despite, motivation, interest, historian, utilizing, rare, usually, cartography, mapmaking, mention, guarantee, usual, cartographic, requirement, exclave, identically, ocean, borders, cartographers, political, northeastern, caribbean, gave, concerning, vassiliev, invariants, obvious, extension, folded, rods, arrange, empty, included, taken, integer, desired, axis, parallel, dimensions, pairs, precise, bounds, earth, interactive, model, mouse, rotate, svg, image, disk, representing, opposite, circle, identified, pentagons, embedding, onto, triple, blobs, ends, tunnels, bubbles, unique, radially, symmetric, wrap, dotted, per, upper, bound, toroidal, polyhedra, sharp, contributions, people, exception, hence, 1934, philip, franklin, 48g, closed, depends, except, outermost, brackets, floor, function, cylinder, sphere, possibly, uncountable, combine, whole, seen, immediate, consequence, simply, expressing, logical, formulae, logic, compactness, kurt, gödel, bruijn, erdős, construction, shows, obtains, therefore, interior, eight, alternate, odd, nevada, missouri, cubic, decide, whether, just, assumptions, consists, multiple, disconnected, disallowing, effect, misconception, directly, numbers, transitive, trick, selected, beforehand, becomes, impossible, exceeding, casual, verifier, think, change, will, simplest, invalid, attempt, forces, true, person, focused, fail, notice, exceeds, replacing, rearranged, been, notorious, attracting, refused, matter, report, fearing, ones, alleged, mentioned, public, scrutiny, decade, refuted, authored, amateurs, times, detail, discussed, immersion, member, eliminate, final, resulting, filled, generated, mechanically, verifying, describing, peer, period, initial, flows, redistributing, preserved, possibilities, positively, charged, enumerating, recall, finally, identify, amenable, reduction, discover, negative, distributed, amongst, internal, fixed, joined, cycle, enumerate, modification, surrounding, recolored, larger, techniques, step, ringed, deal, complicated, rather, subgraphs, specified, described, labelled, demonstrate, somewhere, began, proceeded, exhibit, leaves, mistake, observed, satisfied, run, changing, chains, correctly, say, clockwise, look, necessarily, chained, explore, attached, neighbor, reverse, containing, makes, call, extend, choosing, demonstrates, iv_, abutting, goes, separated, exactly, outer, loss, generality, assume, discussion, introduction, purported, provided, basic, explanation, reworded, formalized, inside, assistant, various, verify, kernel, benjamin, werner, led, shorter, created, improving, similar, reduces, checking, executed, impractical, authors, alternative, quartic, quadratic, 1980s, spread, ulrich, examined, master, thesis, 1981, significant, editor, write, addressing, flaws, replied, due, misinterpretation, results, obliged, claiming, supplement, appeared, explained, corrected, discovered, further, errors, magnum, opus, intelligencer, rwth, aachen, announcement, reported, news, media, postmark, unusual, nature, extensive, verifiable, aroused, considerable, controversy, qualify, genuine, unsatisfying, fell, lacked, sense, structure, answer, appears, kind, monstrous, coincidence, wrote, unlikely, anyone, break, down, something, regard, ordinary, fait, accompli, ian, procedures, properties, infinitude, 834, 482, thousand, hours, independently, daughter, dorothea, blostein, arrangement, contains, condition, either, satisfies, conditions, being, minimum, triangulation, smallest, concepts, teams, racing, algorithmic, during, 1960s, 1970s, mathematician, notably, turned, important, subsequent, expanded, concept, ken, durre, test, unfortunately, juncture, unable, procure, supercomputer, continue, heesch, largest, prime, 1963, celebrated, altering, postage, meter, urbana, champaign, illiac, gillies, formulated, reaching, generalization, unsolved, type, addition, exposing, acclaimed, 1891, unchallenged, julius, percy, peter, alfred, arises, neighborhood, thing, happen, county, principle, inclosure, fully, capable, anything, evident, stand, postulate, failed, believed, followed, didn, derived, facts, guthries, posed, magazine, reference, credits, mine, day, give, him, reason, know, yet, says, figure, compartments, figures, query, necessity, invented, trying, england, brother, former, advisor, inquired, regarding, graduated, became, professor, south, africa, college, letter, oct, william, rowan, hamilton, uses, abstractly, placing, chosen, location, within, corresponds, lead, across, conversely, formed, undirected, corresponding, labeled, belong, want, isolated, otherwise, shape, bizarre, area, infinitely, perimeter, safe, consist, finitely, straight, segments, entirely, surrounds, technically, subset, russia, entire, territory, sufficient, instance, simplified, overseas, territories, kaliningrad, oblast, nakhchivan, autonomous, republic, azerbaijan, cabinda, province, angola, exclaves, enclaves, pie, chart, meaningful, separation, interpreted, appropriately, stated, constructing, adjacencies, denoting, leq, loopless, proceeds, analyzing, improved, managed, decrease, analysis, purpose, software, stronger, significantly, weaker, already, 1800s, resisted, came, mistaken, preceding, decades, means, zero, length, merely, meet, gained, wide, acceptance, reservations, remain, infeasible, ignoring, lakes, oceans, item, projects, printable, download, print, export, switch, legacy, parser, get, shortened, url, cite, information, permanent, link, related, what, actions, talk, tiếng, việt, اردو, українська, türkçe, ไทย, தமிழ், svenska, српски, srpski, shqip, slovenščina, sicilianu, русский, română, português, piemontèis, polski, norsk, nynorsk, nederlands, മലയാളം, latviešu, lietuvių, 한국어, ქართული, 日本語, italiano, ido, bahasa, indonesia, հայերեն, magyar, hrvatski, हिन्दी, עברית, galego, gaeilge, nordfriisk, français, suomi, فارسی, euskara, eesti, español, esperanto, ελληνικά, deutsch, dansk, cymraeg, čeština, کوردی, català, বাংলা, azərbaycanca, asturianu, العربية, personal, special, community, learn, help, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
f the theorem a new approach has led to both a shorter proof and a more efficient algorithm for 4 coloring maps in 1996 neil robertson daniel p sanders paul seymour and robin thomas created a quadratic time algorithm requiring only o n 2 time where n is the number of vertices improving on a quartic time algorithm based on appel and haken s proof 26 the new proof based on the same ideas is similar to appel and haken s but more efficient because it reduces the complexity of the problem and requires checking only 633 reducible configurations both the unavoidability and reducibility parts of this new proof must be executed by a computer and are impractical to check by hand 27 in 2001 the same authors announced an alternative proof by proving the snark conjecture 28 this proof remains unpublished however in 2005 benjamin werner and georges gonthier formalized a proof of the theorem inside the coq proof assistant this removed the need to trust the various computer programs used to verify particular cases it is only necessary to trust the coq kernel 29 summary of proof ideas edit the following discussion is a summary based on the introduction to every planar map is four colorable appel haken 1989 although flawed kempe s original purported proof of the four color theorem provided some of the basic tools later used to prove it the explanation here is reworded in terms of the modern graph theory formulation above kempe s argument goes as follows first if planar regions separated by the graph are not triangulated i e do not have exactly three edges in their boundaries we can add edges without introducing new vertices in order to make every region triangular including the unbounded outer region if this triangulated graph is colorable using four colors or fewer so is the original graph since the same coloring is valid if edges are removed so it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs and without loss of generality we assume the graph is triangulated suppose v e and f are the number of vertices edges and regions faces since each region is triangular and each edge is shared by two regions we have that 2 e 3 f this together with euler s formula v e f 2 can be used to show that 6 v 2 e 12 now the degree of a vertex is the number of edges abutting it if v n is the number of vertices of degree n and d is the maximum degree of any vertex 6 v 2 e 6 i 1 d v i i 1 d i v i i 1 d 6 i v i 12 displaystyle 6v 2e 6 sum _ i 1 d v_ i sum _ i 1 d iv_ i sum _ i 1 d 6 i v_ i 12 but since 12 0 and 6 i 0 for all i 6 this demonstrates that there is at least one vertex of degree 5 or less if there is a graph requiring 5 colors then there is a minimal such graph where removing any vertex makes it four colorable call this graph g then g cannot have a vertex of degree 3 or less because if d v 3 we can remove v from g four color the smaller graph then add back v and extend the four coloring to it by choosing a color different from its neighbors a graph containing a kempe chain consisting of alternating blue and red vertices kempe also showed correctly that g can have no vertex of degree 4 as before we remove the vertex v and four color the remaining vertices if all four neighbors of v are different colors say red green blue and yellow in clockwise order we look for an alternating path of vertices colored red and blue joining the red and blue neighbors such a path is called a kempe chain there may be a kempe chain joining the red and blue neighbors and there may be a kempe chain joining the green and yellow neighbors but not both since these two paths would necessarily intersect and the vertex where they intersect cannot be colored suppose it is the red and blue neighbors that are not chained together explore all vertices attached to the red neighbor by red blue alternating paths and then reverse the colors red and blue on all these vertices the result is still a valid four coloring and v can now be added back and colored red this leaves only the case where g has a vertex of degree 5 but kempe s argument was flawed for this case heawood noticed kempe s mistake and also observed that if one was satisfied with proving only five colors are needed one could run through the above argument changing only that the minimal counterexample requires 6 colors and use kempe chains in the degree 5 situation to prove the five color theorem in any case to deal with this degree 5 vertex case requires a more complicated notion than removing a vertex rather the form of the argument is generalized to considering configurations which are connected subgraphs of g with the degree of each vertex in g specified for example the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in g as above it suffices to demonstrate that if the configuration is removed and the remaining graph four colored then the coloring can be modified in such a way that when the configuration is re added the four coloring can be extended to it as well a configuration for which this is possible is called a reducible configuration if at least one of a set of configurations must occur somewhere in g that set is called unavoidable the argument above began by giving an unavoidable set of five configurations a single vertex with degree 1 a single vertex with degree 2 a single vertex with degree 5 and then proceeded to show that the first 4 are reducible to exhibit an unavoidable set of configurations where every configuration in the set is reducible would prove the theorem because g is triangular the degree of each vertex in a configuration is known and all edges internal to the configuration are known the number of vertices in g adjacent to a given configuration is fixed and they are joined in a cycle these vertices form the ring of the configuration a configuration with k vertices in its ring is a k ring configuration and the configuration together with its ring is called the ringed configuration as in the simple cases above one may enumerate all distinct four colorings of the ring any coloring that can be extended without modification to a coloring of the configuration is called initially good for example the single vertex configuration above with 3 or fewer neighbors were initially good in general the surrounding graph must be systematically recolored to turn the ring s coloring into a good one as was done in the case above where there were 4 neighbors for a general configuration with a larger ring this requires more complex techniques because of the large number of distinct four colorings of the ring this is the primary step requiring computer assistance finally it remains to identify an unavoidable set of configurations amenable to reduction by this procedure the primary method used to discover such a set is the method of discharging the intuitive idea underlying discharging is to consider the planar graph as an electrical network initially positive and negative electrical charge is distributed amongst the vertices so that the total is positive recall the formula above i 1 d 6 i v i 12 displaystyle sum _ i 1 d 6 i v_ i 12 each vertex with degree i displaystyle i is assigned an initial charge of 6 i displaystyle 6 i then one flows the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules the discharging procedure since the total charge was initially positive 12 and charge is preserved some vertices still have positive charge the rules restrict the possibilities for configurations of positively charged vertices so enumerating all such possible configurations gives an unavoidable set as long as some member of the unavoidable set is not reducible the discharging procedure is modified to eliminate it while introducing other configurations appel and haken s final discharging procedure was extremely complex and together with a description of the resulting unavoidable configuration set filled a 400 page volume but the configurations it generated could be checked mechanically to be reducible verifying the volume describing the unavoidable configuration set itself was done by peer review over a period of several years a technical detail not discussed here but required to complete the proof is immersion reducibility false disproofs edit the four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history at first the new york times refused as a matter of policy to report on the appel haken proof fearing that the proof would be shown false like the ones before it 21 some alleged proofs like kempe s and tait s mentioned above stood under public scrutiny for over a decade before they were refuted but many more authored by amateurs were never published in the first map which exceeds four colors replacing the red regions with any of the four other colors would not work and the example may initially appear to violate the theorem however the colors can be rearranged as in the second map generally the simplest though invalid counterexamples attempt to create one region which touches all other regions this forces the remaining regions to be colored with only three colors because the four color theorem is true this is always possible however because the person drawing the map is focused on the one large region they fail to notice that the remaining regions can in fact be colored with three colors this trick can be generalized there are many maps where if the colors of some regions are selected beforehand it becomes impossible to color the remaining regions without exceeding four colors a casual verifier of the counterexample may not think to change the colors of these regions so that the counterexample will appear as though it is valid perhaps one effect underlying this common misconception is the fact that the color restriction is not transitive a region only has to be colored differently from regions it touches directly not regions touching regions that it touches if this were the restriction planar graphs would require arbitrarily large numbers of colors other false disproofs violate the assumptions of the theorem such as using a region that consists of multiple disconnected parts or disallowing regions of the same color from touching at a point three coloring edit proof without words that a map of us states needs at least four colors while every planar map can be colored with four colors it is np complete in complexity to decide whether an arbitrary planar map can be colored with just three colors 30 a cubic map can be colored with only three colors if and only if each interior region has an even number of neighboring regions 31 in the us states map example landlocked missouri mo has eight neighbors an even number it must be differently colored from all of them but the neighbors can alternate colors thus this part of the map needs only three colors however landlocked nevada nv has five neighbors an odd number these neighbors require three colors and it must be differently colored from them thus four colors are needed here generalizations edit infinite graphs edit by joining the single arrows together and the double arrows together one obtains a torus with seven mutually touching regions therefore seven colors are necessary this construction shows the torus divided into the maximum of seven regions each one of which touches every other the four color theorem applies not only to finite planar graphs but also to infinite graphs that can be drawn without crossings in the plane and even more generally to infinite graphs possibly with an uncountable number of vertices for which every finite subgraph is planar to prove this one can combine a proof of the theorem for finite planar graphs with the de bruijn erdős theorem stating that if every finite subgraph of an infinite graph is k colorable then the whole graph is also k colorable nash williams 1967 this can also be seen as an immediate consequence of kurt gödel s compactness theorem for first order logic simply by expressing the colorability of an infinite graph with a set of logical formulae higher surfaces edit one can also consider the coloring problem on surfaces other than the plane 32 the problem on the sphere or cylinder is equivalent to that on the plane for closed orientable or non orientable surfaces with positive genus the maximum number p of colors needed depends on the euler characteristic χ of the surface except for the klein bottle the formula is as follows p 7 49 24 χ 2 displaystyle p left lfloor frac 7 sqrt 49 24 chi 2 right rfloor where the outermost brackets denote the floor function for an orientable surface this implies that p can be given in terms of the genus g of the surface p 7 1 48 g 2 displaystyle p left lfloor frac 7 sqrt 1 48g 2 right rfloor the top formula the heawood conjecture was proposed by p j heawood in 1890 and after contributions by several people proved by gerhard ringel and j w t youngs in 1968 the only exception to the formula is the klein bottle which has euler characteristic χ 0 hence the formula gives p 7 but requires only six colors as shown by philip franklin in 1934 for example the torus has euler characteristic χ 0 and genus g 1 and thus p 7 so no more than seven colors are required to color any map on a torus this upper bound of 7 is sharp certain toroidal polyhedra such as the szilassi polyhedron require seven colors a möbius strip requires six colors tietze 1910 as do 1 planar graphs graphs drawn with at most one simple crossing per edge borodin 1984 if both the vertices and the faces of a planar graph are colored in such a way that no two adjacent vertices faces or vertex face pair have the same color then again at most six colors are needed borodin 1984 the real projective plane has euler characteristic χ 1 and the formula gives p 6 so no more than six colors are required a radially symmetric 7 colored torus regions of the same color wrap around along dotted lines an 8 colored double torus genus two surface bubbles denote unique contact of two regions a 9 colored triple torus genus three surface blobs denote ends of their respective tunnels a 6 colored klein bottle tietze s subdivision of a möbius strip into six mutually adjacent regions requiring six colors the vertices and edges of the subdivision form an embedding of tietze s graph onto the strip a disk representing the real projective plane opposite points on the circle are identified the projective plane can be divided into six pentagons based on the petersen graph giving a 6 coloring interactive szilassi polyhedron model each of the seven faces is adjacent to every other in the svg image move the mouse to rotate it for graphs whose vertices are represented as pairs of points on two distinct surfaces with edges drawn as non crossing curves on one of the two surfaces the chromatic number can be at least 9 and is at most 12 but...