1140:
1444:
1437:
29:
1451:
1610:
1603:
1596:
1589:
1534:
1527:
1520:
1513:
1458:
537:
311:
243:
175:
398:
1020:
937:
107:
256:
2000:
1318:
into three-dimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any three-dimensional embedding of
188:
120:
532:{\displaystyle \left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.}
1676:
1909:
1696:
1381:
1361:
2132:
1109:
requiring either 7233 or 7234 crossings. Further values are collected by the
Rectilinear Crossing Number project. Rectilinear Crossing numbers for
1120:
0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180, ... (sequence
1876:
1246:. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph
2074:
1720:
Bang-Jensen, Jørgen; Gutin, Gregory (2018), "Basic
Terminology, Notation and Results", in Bang-Jensen, Jørgen; Gutin, Gregory (eds.),
1127:
1043:
1992:
2341:
956:
606:
869:
1762:
2194:
1868:
1851:
66:
2039:
2155:
2136:
1036:
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, ... (sequence
2277:
2209:
1820:
1088:
1030:
657:
2217:
2213:
762:
649:
in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).
306:{\displaystyle \left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.}
1148:
781:
2176:
1843:
1837:
947:
789:
2053:
1624:
1630:
1269:
238:{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}
170:{\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}
2346:
2282:
1256:
1198:
1026:
566:
181:
113:
1641:
where every vertex on one side of the bipartition is connected to every vertex on the other side
2048:
576:
1810:
1734:
758:
748:
634:
571:
391:
47:
1750:
2251:
2182:
2113:
2031:
716:
249:
8:
1655:
1281:
1219:
1215:
848:
638:
59:
2186:
2117:
1724:, Springer Monographs in Mathematics, Springer International Publishing, pp. 1–34,
2255:
2229:
2103:
1961:
1901:
1681:
1366:
1346:
1332:
1306:
1144:
317:
2314:
2190:
2066:
1905:
1893:
1847:
1816:
1806:
1758:
1328:
740:
2243:
2020:
Hassani, M. "Cycles in graphs and derangements." Math. Gaz. 88, 123–126, 2004.
2317:
2291:
2259:
2239:
2058:
1971:
1885:
1725:
1074:
1059:
770:
630:
407:
343:
2247:
2159:
1638:
1302:
1289:
665:
561:
355:
327:
1729:
2172:
1976:
1949:
785:
653:
646:
581:
1139:
265:
197:
129:
2335:
2062:
1897:
744:
661:
549:
2295:
2273:
2070:
1746:
1243:
1051:
627:
619:
1802:
1285:
1194:
774:
669:
615:
715:, and other sources state that the notation honors the contributions of
2150:
1889:
766:
2032:"Extremal problems for topological indices in combinatorial chemistry"
1867:
Joos, Felix; Kim, Jaehoon; Kühn, Daniela; Osthus, Deryk (2019-08-05).
1309:. In other words, and as Conway and Gordon proved, every embedding of
2322:
2234:
1699:
1450:
1443:
1436:
28:
1966:
1609:
1602:
1383:
between 1 and 12, are shown below along with the numbers of edges:
1181:
2108:
1595:
1588:
1533:
1526:
1519:
1512:
1457:
664:
of complete graphs, with their vertices placed on the points of a
1645:
1168:
769:
which disconnects the graph is the complete set of vertices. The
1948:
Montgomery, Richard; Pokrovskiy, Alexey; Sudakov, Benny (2021).
1778:
1255:
plays a key role in the characterizations of planar graphs: by
2100:
A combinatorial survey of identities for the double factorial
1202:
2272:
1947:
1122:
1038:
668:, had already appeared in the 13th century, in the work of
526:
300:
232:
164:
1222:
in four or more dimensions also has a complete skeleton.
823:
vertices. Ringel's conjecture asks if the complete graph
652:
Graph theory itself is typically dated as beginning with
2312:
1259:, a graph is planar if and only if it contains neither
840:
edges. This is known to be true for sufficiently large
2208:
70:
1684:
1658:
1369:
1349:
1050:
These numbers give the largest possible value of the
1015:{\displaystyle e_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}.}
959:
872:
401:
259:
191:
123:
69:
2220:(1993), "Linkless embeddings of graphs in 3-space",
932:{\displaystyle w_{n+2}=n!e_{n}=\lfloor en!\rfloor ,}
780:If the edges of a complete graph are each given an
1690:
1670:
1375:
1355:
1014:
931:
531:
305:
237:
169:
101:
1866:
2333:
2178:Time Travel and Other Mathematical Bewilderments
1719:
1201:, a nonconvex polyhedron with the topology of a
1993:"Rainbow Proof Shows Graphs Have Uniform Parts"
836:can be decomposed into copies of any tree with
672:. Such a drawing is sometimes referred to as a
2162:, Bolyai Institute, University of Szeged, 1949
1801:
102:{\displaystyle \textstyle {\frac {n(n-1)}{2}}}
16:Graph in which every two vertices are adjacent
2280:(1983). "Knots and Links in Spatial Graphs".
2222:Bulletin of the American Mathematical Society
1753:, in Wilson, Robin; Watkins, John J. (eds.),
2130:
1877:Journal of the European Mathematical Society
923:
911:
706:
705:, but the German name for a complete graph,
700:
699:in this notation stands for the German word
2029:
1147:model with vertices representing nodes. In
2181:, W. H. Freeman and Company, p. 140,
2030:Tichy, Robert F.; Wagner, Stephan (2005),
1869:"Optimal packings of bounded degree trees"
1757:, Oxford University Press, pp. 7–37,
1288:in place of subdivisions. As part of the
2233:
2107:
2052:
1975:
1965:
1937:. Proceedings of the Symposium Smolenice.
1138:
1134:
1029:of the complete graphs are given by the
2171:
851:between a specific pair of vertices in
2334:
2097:
1932:
1835:
2313:
2133:"Rectilinear Crossing Number project"
1935:Theory of Graphs and its Applications
1751:"Two thousand years of combinatorics"
1745:
757:. All complete graphs are their own
695:. Some sources claim that the letter
1990:
1812:A Logical Approach to Discrete Math
1301:plays a similar role as one of the
13:
410:
268:
41:, a complete graph with 7 vertices
14:
2358:
2306:
1954:Geometric and Functional Analysis
1755:Combinatorics: Ancient and Modern
1159:nodes represents the edges of an
2040:Journal of Computational Biology
1950:"A proof of Ringel's Conjecture"
1815:, Springer-Verlag, p. 436,
1608:
1601:
1594:
1587:
1532:
1525:
1518:
1511:
1456:
1449:
1442:
1435:
633:in which every pair of distinct
27:
2266:
2244:10.1090/S0273-0979-1993-00335-5
2202:
2165:
2152:A Polyhedron Without Diagonals.
2143:
2124:
2091:
2080:from the original on 2017-09-21
2023:
2014:
2003:from the original on 2020-02-20
1915:from the original on 2020-03-09
1331:that is embedded in space as a
1984:
1941:
1926:
1860:
1829:
1795:
1771:
1739:
1713:
1151:, move the mouse to rotate it.
711:, does not contain the letter
607:Table of graphs and parameters
479:
466:
89:
77:
1:
2342:Parametric families of graphs
1706:
1058:-vertex graph. The number of
679:
7:
1730:10.1007/978-3-319-71840-8_1
1618:
1338:
847:The number of all distinct
658:Seven Bridges of Königsberg
10:
2363:
1977:10.1007/s00039-021-00576-2
1842:, Addison Wesley, p.
1836:Pirnot, Thomas L. (2000),
1722:Classes of Directed Graphs
1648:, which is identical to a
1284:the same result holds for
773:of a complete graph is an
1280:as a subdivision, and by
1205:, has the complete graph
637:is connected by a unique
605:
590:
542:
390:
354:
342:
316:
248:
180:
112:
58:
46:
26:
21:
2063:10.1089/cmb.2005.12.1004
1631:Complete bipartite graph
1627:, in computer networking
1270:complete bipartite graph
1180:forms the edge set of a
2283:Journal of Graph Theory
1625:Fully connected network
801:can be decomposed into
688:vertices is denoted by
2296:10.1002/jgt.3190070410
2098:Callan, David (2009),
1839:Mathematics All Around
1692:
1672:
1377:
1357:
1155:A complete graph with
1152:
1073:even) is given by the
1062:of the complete graph
1016:
993:
933:
707:
701:
684:The complete graph on
533:
307:
239:
171:
103:
1693:
1673:
1378:
1358:
1142:
1135:Geometry and topology
1017:
973:
934:
761:. They are maximally
534:
308:
240:
172:
104:
1682:
1656:
1367:
1347:
1257:Kuratowski's theorem
957:
870:
717:Kazimierz Kuratowski
656:'s 1736 work on the
399:
257:
189:
121:
67:
2187:1988ttom.book.....G
2118:2009arXiv0906.1317C
1933:Ringel, G. (1963).
1671:{\displaystyle n+1}
1343:Complete graphs on
1220:neighborly polytope
708:vollständiger Graph
2315:Weisstein, Eric W.
2158:2017-09-18 at the
2131:Oswin Aichholzer.
1807:Schneider, Fred B.
1688:
1668:
1373:
1353:
1307:linkless embedding
1199:Császár polyhedron
1153:
1145:Csaszar polyhedron
1012:
929:
529:
524:
303:
298:
235:
230:
167:
162:
99:
98:
1991:Hartnett, Kevin.
1884:(12): 3573–3647.
1783:, nrich.maths.org
1691:{\displaystyle n}
1616:
1615:
1376:{\displaystyle n}
1356:{\displaystyle n}
1329:Hamiltonian cycle
1171:. Geometrically
1060:perfect matchings
1031:telephone numbers
1007:
741:triangular number
719:to graph theory.
612:
611:
567:Vertex-transitive
520:
294:
226:
158:
96:
2354:
2328:
2327:
2318:"Complete Graph"
2300:
2299:
2270:
2264:
2262:
2237:
2206:
2200:
2199:
2169:
2163:
2147:
2141:
2140:
2135:. Archived from
2128:
2122:
2120:
2111:
2095:
2089:
2087:
2086:
2085:
2079:
2056:
2047:(7): 1004–1013,
2036:
2027:
2021:
2018:
2012:
2011:
2009:
2008:
1988:
1982:
1981:
1979:
1969:
1945:
1939:
1938:
1930:
1924:
1923:
1921:
1920:
1914:
1890:10.4171/JEMS/909
1873:
1864:
1858:
1856:
1833:
1827:
1825:
1799:
1793:
1791:
1790:
1788:
1775:
1769:
1767:
1747:Knuth, Donald E.
1743:
1737:
1732:
1717:
1697:
1695:
1694:
1689:
1678:vertices, where
1677:
1675:
1674:
1669:
1612:
1605:
1598:
1591:
1582:
1571:
1560:
1549:
1536:
1529:
1522:
1515:
1506:
1495:
1484:
1473:
1460:
1453:
1446:
1439:
1430:
1419:
1408:
1397:
1386:
1385:
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1326:
1317:
1303:forbidden minors
1300:
1282:Wagner's theorem
1279:
1267:
1254:
1241:
1232:
1213:
1192:
1179:
1166:
1158:
1125:
1115:
1108:
1100:are known, with
1099:
1089:crossing numbers
1083:
1075:double factorial
1072:
1068:
1057:
1041:
1021:
1019:
1018:
1013:
1008:
1006:
995:
992:
987:
969:
968:
948:Euler's constant
945:
938:
936:
935:
930:
907:
906:
888:
887:
862:
843:
839:
835:
822:
818:
811:
804:
800:
784:, the resulting
771:complement graph
756:
738:
727:
714:
710:
704:
698:
694:
687:
643:complete digraph
631:undirected graph
601:
577:Strongly regular
556:
538:
536:
535:
530:
528:
525:
521:
518:
514:
510:
509:
508:
487:
486:
445:
441:
440:
382:
378:
369:
365:
350:
344:Chromatic number
338:
312:
310:
309:
304:
302:
299:
295:
292:
244:
242:
241:
236:
234:
231:
227:
224:
176:
174:
173:
168:
166:
163:
159:
156:
108:
106:
105:
100:
97:
92:
72:
54:
40:
31:
19:
18:
2362:
2361:
2357:
2356:
2355:
2353:
2352:
2351:
2332:
2331:
2309:
2304:
2303:
2271:
2267:
2210:Robertson, Neil
2207:
2203:
2197:
2173:Gardner, Martin
2170:
2166:
2160:Wayback Machine
2148:
2144:
2129:
2125:
2096:
2092:
2083:
2081:
2077:
2054:10.1.1.379.8693
2034:
2028:
2024:
2019:
2015:
2006:
2004:
1997:Quanta Magazine
1989:
1985:
1946:
1942:
1931:
1927:
1918:
1916:
1912:
1871:
1865:
1861:
1854:
1834:
1830:
1823:
1800:
1796:
1786:
1784:
1777:
1776:
1772:
1765:
1744:
1740:
1718:
1714:
1709:
1702:of the simplex.
1683:
1680:
1679:
1657:
1654:
1653:
1639:bipartite graph
1621:
1580:
1574:
1569:
1563:
1558:
1552:
1547:
1541:
1504:
1498:
1493:
1487:
1482:
1476:
1471:
1465:
1428:
1422:
1417:
1411:
1406:
1400:
1395:
1389:
1368:
1365:
1364:
1348:
1345:
1344:
1341:
1333:nontrivial knot
1325:
1319:
1316:
1310:
1299:
1293:
1290:Petersen family
1278:
1272:
1266:
1260:
1253:
1247:
1240:
1234:
1231:
1225:
1212:
1206:
1191:
1185:
1178:
1172:
1160:
1156:
1137:
1121:
1114:
1110:
1107:
1101:
1098:
1092:
1077:
1070:
1067:
1063:
1055:
1037:
999:
994:
988:
977:
964:
960:
958:
955:
954:
943:
902:
898:
877:
873:
871:
868:
867:
861:
852:
841:
837:
834:
824:
820:
817:
813:
810:
806:
802:
799:
795:
759:maximal cliques
751:
729:
726:
722:
712:
696:
693:
689:
685:
682:
666:regular polygon
599:
594:
586:
572:Edge-transitive
562:Symmetric graph
550:
523:
522:
517:
515:
498:
494:
482:
478:
465:
461:
458:
457:
446:
436:
432:
428:
425:
424:
413:
406:
402:
400:
397:
396:
386:
380:
373:
367:
363:
356:Chromatic index
348:
336:
322:
297:
296:
291:
289:
283:
282:
271:
264:
260:
258:
255:
254:
229:
228:
223:
221:
215:
214:
203:
196:
192:
190:
187:
186:
161:
160:
155:
153:
147:
146:
135:
128:
124:
122:
119:
118:
73:
71:
68:
65:
64:
52:
42:
39:
33:
17:
12:
11:
5:
2360:
2350:
2349:
2347:Regular graphs
2344:
2330:
2329:
2308:
2307:External links
2305:
2302:
2301:
2290:(4): 445–453.
2278:Cameron Gordon
2265:
2214:Seymour, P. D.
2201:
2195:
2164:
2149:Ákos Császár,
2142:
2139:on 2007-04-30.
2123:
2090:
2022:
2013:
1983:
1960:(3): 663–720.
1940:
1925:
1859:
1852:
1828:
1821:
1794:
1770:
1764:978-0191630620
1763:
1738:
1711:
1710:
1708:
1705:
1704:
1703:
1687:
1667:
1664:
1661:
1650:complete graph
1642:
1628:
1620:
1617:
1614:
1613:
1606:
1599:
1592:
1584:
1583:
1578:
1572:
1567:
1561:
1556:
1550:
1545:
1538:
1537:
1530:
1523:
1516:
1508:
1507:
1502:
1496:
1491:
1485:
1480:
1474:
1469:
1462:
1461:
1454:
1447:
1440:
1432:
1431:
1426:
1420:
1415:
1409:
1404:
1398:
1393:
1372:
1363:vertices, for
1352:
1340:
1337:
1323:
1314:
1297:
1276:
1264:
1251:
1238:
1229:
1210:
1189:
1176:
1136:
1133:
1132:
1131:
1112:
1105:
1096:
1065:
1048:
1047:
1025:The number of
1023:
1022:
1011:
1005:
1002:
998:
991:
986:
983:
980:
976:
972:
967:
963:
940:
939:
928:
925:
922:
919:
916:
913:
910:
905:
901:
897:
894:
891:
886:
883:
880:
876:
856:
828:
815:
808:
797:
786:directed graph
724:
691:
681:
678:
654:Leonhard Euler
647:directed graph
624:complete graph
610:
609:
603:
602:
597:
592:
588:
587:
585:
584:
579:
574:
569:
564:
559:
546:
544:
540:
539:
527:
516:
513:
507:
504:
501:
497:
493:
490:
485:
481:
477:
474:
471:
468:
464:
460:
459:
456:
453:
450:
447:
444:
439:
435:
431:
427:
426:
423:
420:
417:
414:
412:
409:
408:
405:
394:
388:
387:
385:
384:
371:
360:
358:
352:
351:
346:
340:
339:
332:
320:
314:
313:
301:
290:
288:
285:
284:
281:
278:
275:
272:
270:
267:
266:
263:
252:
246:
245:
233:
222:
220:
217:
216:
213:
210:
207:
204:
202:
199:
198:
195:
184:
178:
177:
165:
154:
152:
149:
148:
145:
142:
139:
136:
134:
131:
130:
127:
116:
110:
109:
95:
91:
88:
85:
82:
79:
76:
62:
56:
55:
50:
44:
43:
37:
32:
24:
23:
22:Complete graph
15:
9:
6:
4:
3:
2:
2359:
2348:
2345:
2343:
2340:
2339:
2337:
2325:
2324:
2319:
2316:
2311:
2310:
2297:
2293:
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670:Ramon Llull
660:. However,
2336:Categories
2084:2012-03-29
2007:2020-02-20
1967:2001.02665
1919:2020-03-09
1822:0387941150
1787:23 January
1707:References
946:refers to
812:such that
790:tournament
767:vertex cut
680:Properties
543:Properties
2323:MathWorld
2109:0906.1317
2049:CiteSeerX
1906:119315954
1898:1435-9855
1700:dimension
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975:∑
924:⌋
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2156:Archived
2075:archived
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1268:nor the
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635:vertices
591:Notation
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557:-regular
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2260:1110662
2252:1164063
2183:Bibcode
2114:Bibcode
1698:is the
1646:simplex
1214:as its
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1126:in the
1123:A014540
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2256:S2CID
2230:arXiv
2104:arXiv
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1962:arXiv
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