1358:
29:
436:
1618:
operations. Specifically, such a sequence can begin by forming each of the independent sets of the Turán graph as a disjoint union of isolated vertices. Then, the overall graph is the complement of the disjoint union of the complements of these independent sets.
234:
320:
1344:
extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the
985:
885:
333:
140:
1471:
couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason, it is also called the
1338:
153:
782:
247:
1568:
1647:
develop an efficient algorithm for finding clusters of orthologous groups of genes in genome data, by representing the data as a graph and searching for large Turán subgraphs.
915:
719:
556:
505:
1514:, although some authors consider Turán graphs to be a trivial case of strong regularity and therefore exclude them from the definition of a strongly regular graph.
1109:
1590: ≤ 2; each maximal clique is formed by choosing one vertex from each partition subset. This is the largest number of maximal cliques possible among all
1205:
1135:
1057:
1031:
1179:
1159:
1077:
1005:
920:
684:
664:
644:
624:
604:
584:
464:
71:
790:
1236: + 1 vertices in the Turán graph includes two vertices in the same partition subset; therefore, the Turán graph does not contain a
606:
subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to different subsets. Where
431:{\displaystyle \left\{{\begin{array}{ll}\infty &r=1\vee (n\leq 3\wedge r\leq 2)\\4&r=2\\3&{\text{otherwise}}\end{array}}\right.}
85:
1994:
1885:
2154:
1271:
511:
229:{\displaystyle \left\{{\begin{array}{ll}\infty &r=1\\2&r\leq n/2\\1&{\text{otherwise}}\end{array}}\right.}
1244: + 1. According to Turán's theorem, the Turán graph has the maximum possible number of edges among all (
315:{\displaystyle \left\{{\begin{array}{ll}\infty &r=1\\1&r=n\\2&{\text{otherwise}}\end{array}}\right.}
2159:
1955:
1841:
2051:
1920:
1690:
724:
1678:, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex.
1524:
2049:
Witsenhausen, H. S. (1974). "On the maximum of the sum of squared distances under a diameter constraint".
559:
1517:
The class of Turán graphs can have exponentially many maximal cliques, meaning this class does not have
1989:
1341:
1615:
1487:
1594:-vertex graphs regardless of the number of edges in the graph; these graphs are sometimes called
1518:
240:
146:
1717:
colors. The partition of the Turán graph into independent sets corresponds to the partition of
1651:
1511:
1237:
1226:
894:
49:
1216:
689:
526:
475:
2036:(1941). "Egy gráfelméleti szélsőértékfeladatról (On an extremal problem in graph theory)".
1877:
1630:
1082:
326:
980:{\displaystyle \left\lfloor \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}\right\rfloor }
8:
1437:
1184:
1114:
1036:
1010:
77:
2092:
2068:
1964:
1937:
1902:
1710:
1164:
1144:
1062:
990:
669:
649:
629:
609:
589:
569:
563:
449:
56:
1380:(6,3). Unconnected vertices are given the same color in this face-centered projection.
1357:
2127:
2111:
2108:
2089:
1855:
1836:
1906:
2060:
2007:
1999:
1974:
1941:
1929:
1894:
1865:
1850:
1460:
1404:
1346:
880:{\displaystyle \left(1-{\frac {1}{r}}\right){\frac {n^{2}-s^{2}}{2}}+{s \choose 2}}
442:
2021:
1998:. Lecture Notes in Computer Science no. 3383, Springer-Verlag. pp. 395–402.
1507:
2130:
1633:. Nikiforov (2005) uses Turán graphs to supply a lower bound for the sum of the
2033:
1979:
1950:
1611:
1571:
1445:
1408:
1366:
1222:
42:
1898:
1388:
in a Turán graph lead to notable graphs that have been independently studied.
342:
256:
162:
2148:
1663:
1138:
1721:
into color classes. In particular, the Turán graph is the unique maximal
1495:
1969:
2072:
2012:
1933:
1638:
1464:
1362:
2024:(1969). Tutte, W.T. (ed.). "On the boxicity and cubicity of a graph".
1610:; that is, it can be formed from individual vertices by a sequence of
2135:
2116:
2097:
1915:
2064:
2003:
1425:
1607:
1225:, who used them to prove Turán's theorem, an important result in
135:{\displaystyle \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}}
1787:
1650:
Turán graphs also have some interesting properties related to
2087:
1670:
conjectures that the maximum sum of squared distances, among
28:
425:
309:
223:
2106:
1739:
1866:"Computing high-stringency COGs using Turán type graphs"
1333:{\displaystyle {\frac {r\,{-}\,1}{3r}}(2\alpha -1)n^{2}}
1864:
Falls, Craig; Powell, Bradford; Snoeyink, Jack (2003).
1811:
1775:
1527:
1274:
1187:
1167:
1147:
1117:
1085:
1065:
1039:
1013:
993:
923:
897:
793:
727:
692:
672:
652:
632:
612:
592:
572:
529:
478:
452:
336:
250:
156:
88:
59:
2125:
1763:
1863:
1793:
1644:
917:, this edge count can be more succinctly stated as
1878:"Local density in graphs with forbidden subgraphs"
1799:
1751:
1562:
1332:
1199:
1173:
1153:
1129:
1103:
1071:
1051:
1025:
999:
979:
909:
879:
776:
713:
678:
658:
638:
618:
598:
578:
550:
499:
458:
430:
314:
228:
134:
65:
871:
858:
2146:
1875:
1745:
1253:
1951:"Eigenvalue problems of Nordhaus-Gaddum type"
2048:
1817:
1667:
1554:
1540:
1340:edges, if α is sufficiently close to 1. The
1256:show that the Turán graph is also the only (
1834:
1781:
1622:
1260: + 1)-clique-free graph of order
646:are the quotient and remainder of dividing
1662:)) on the volume of any three-dimensional
1232:By the pigeonhole principle, every set of
2011:
1978:
1968:
1948:
1913:
1854:
1769:
1287:
1281:
1995:Proc. Int. Symp. Graph Drawing (GD 2004)
1886:Combinatorics, Probability and Computing
1356:
1248: + 1)-clique-free graphs with
2020:
1987:
1876:Keevash, Peter; Sudakov, Benny (2003).
1805:
1757:
1655:
1421:
777:{\displaystyle K_{q+1,q+1,\ldots ,q,q}}
2147:
1563:{\displaystyle T(n,\lceil n/3\rceil )}
2126:
2107:
2088:
2032:
1835:Chao, C. Y.; Novacky, G. A. (1982).
16:Balanced complete multipartite graph
1794:Falls, Powell & Snoeyink (2003)
1645:Falls, Powell & Snoeyink (2003)
1601:
13:
1992:(2005). "No-three-in-line-in-3D".
1210:
862:
345:
259:
165:
14:
2171:
2081:
1384:Several choices of the parameter
2026:Recent Progress in Combinatorics
1918:(1965). "On cliques in graphs".
1629:: no other graphs have the same
1352:
27:
1837:"On maximally saturated graphs"
1641:of a graph and its complement.
1625:show that the Turán graphs are
1521:. For example, the Turán graph
1432:; it is sometimes known as the
1264:in which every subset of α
1557:
1531:
1403:) can be formed by removing a
1369:whose edges and vertices form
1317:
1302:
545:
533:
512:Table of graphs and parameters
494:
482:
386:
362:
1:
2155:Parametric families of graphs
2052:American Mathematical Monthly
1921:Israel Journal of Mathematics
1827:
1674:points with unit diameter in
1221:Turán graphs are named after
784:, and the number of edges is
2038:Matematikai és Fizikai Lapok
1949:Nikiforov, Vladimir (2007).
1856:10.1016/0012-365X(82)90200-X
1746:Keevash & Sudakov (2003)
1254:Keevash & Sudakov (2003)
721:), the graph is of the form
7:
1729:-color equitable coloring.
1463:, the graph of the regular
1436:. This graph is also the 1-
560:complete multipartite graph
10:
2176:
1980:10.1016/j.disc.2006.07.035
1448:; for instance, the graph
1214:
1899:10.1017/S0963548302005539
1782:Chao & Novacky (1982)
1658:give a lower bound of Ω((
1623:Chao & Novacky (1982)
1079:; each vertex has degree
510:
469:
441:
325:
239:
145:
76:
48:
38:
26:
21:
1732:
1488:complete bipartite graph
1268:vertices spans at least
1770:Moon & Moser (1965)
1606:Every Turán graph is a
1424:showed, this graph has
910:{\displaystyle r\leq 7}
33:The Turán graph T(13,4)
2093:"Cocktail Party Graph"
1725:-vertex graph with an
1652:geometric graph theory
1564:
1381:
1334:
1201:
1175:
1155:
1131:
1105:
1073:
1053:
1027:
1001:
981:
911:
881:
778:
715:
714:{\displaystyle n=qr+s}
680:
660:
640:
620:
600:
580:
552:
551:{\displaystyle T(n,r)}
501:
500:{\displaystyle T(n,r)}
460:
432:
316:
230:
136:
67:
2160:Extremal graph theory
1806:Pór & Wood (2005)
1656:Pór & Wood (2005)
1631:chromatic polynomials
1565:
1506:, the Turán graph is
1360:
1335:
1227:extremal graph theory
1202:
1176:
1156:
1132:
1106:
1104:{\displaystyle n-q-1}
1074:
1054:
1028:
1002:
982:
912:
882:
779:
716:
681:
661:
641:
621:
601:
581:
553:
502:
461:
433:
317:
231:
137:
68:
1956:Discrete Mathematics
1842:Discrete Mathematics
1666:of the Turán graph.
1627:chromatically unique
1525:
1473:cocktail party graph
1272:
1185:
1165:
1145:
1115:
1083:
1063:
1037:
1011:
991:
921:
895:
791:
725:
690:
670:
650:
630:
610:
590:
570:
527:
476:
450:
334:
248:
154:
86:
57:
1818:Witsenhausen (1974)
1668:Witsenhausen (1974)
1342:Erdős–Stone theorem
1200:{\displaystyle s=0}
1130:{\displaystyle n-q}
1052:{\displaystyle r-s}
1026:{\displaystyle q+1}
2128:Weisstein, Eric W.
2112:"Octahedral Graph"
2109:Weisstein, Eric W.
2090:Weisstein, Eric W.
1934:10.1007/BF02760024
1711:equitable coloring
1560:
1452:(6,3) =
1382:
1330:
1197:
1171:
1151:
1127:
1101:
1069:
1049:
1023:
997:
977:
907:
877:
774:
711:
676:
656:
636:
616:
596:
576:
564:partitioning a set
562:; it is formed by
548:
497:
456:
428:
423:
312:
307:
226:
221:
132:
63:
2059:(10): 1100–1101.
1705:) if and only if
1693:of a Turán graph
1596:Moon–Moser graphs
1349:of the subgraph.
1300:
1174:{\displaystyle r}
1154:{\displaystyle n}
1072:{\displaystyle q}
1000:{\displaystyle s}
970:
948:
869:
850:
813:
679:{\displaystyle r}
659:{\displaystyle n}
639:{\displaystyle s}
619:{\displaystyle q}
599:{\displaystyle r}
579:{\displaystyle n}
517:
516:
459:{\displaystyle r}
419:
303:
217:
130:
108:
66:{\displaystyle n}
2167:
2141:
2140:
2122:
2121:
2103:
2102:
2076:
2045:
2029:
2017:
2015:
1984:
1982:
1972:
1945:
1910:
1882:
1872:
1870:
1860:
1858:
1821:
1815:
1809:
1803:
1797:
1791:
1785:
1779:
1773:
1767:
1761:
1755:
1749:
1743:
1602:Other properties
1569:
1567:
1566:
1561:
1550:
1512:strongly regular
1502:is a divisor of
1478:The Turán graph
1461:octahedral graph
1405:perfect matching
1391:The Turán graph
1376:, a Turán graph
1347:chromatic number
1339:
1337:
1336:
1331:
1329:
1328:
1301:
1299:
1291:
1286:
1276:
1252: vertices.
1206:
1204:
1203:
1198:
1180:
1178:
1177:
1172:
1161:is divisible by
1160:
1158:
1157:
1152:
1136:
1134:
1133:
1128:
1110:
1108:
1107:
1102:
1078:
1076:
1075:
1070:
1059:subsets of size
1058:
1056:
1055:
1050:
1032:
1030:
1029:
1024:
1007:subsets of size
1006:
1004:
1003:
998:
987:. The graph has
986:
984:
983:
978:
976:
972:
971:
966:
965:
956:
954:
950:
949:
941:
916:
914:
913:
908:
886:
884:
883:
878:
876:
875:
874:
861:
851:
846:
845:
844:
832:
831:
821:
819:
815:
814:
806:
783:
781:
780:
775:
773:
772:
720:
718:
717:
712:
685:
683:
682:
677:
665:
663:
662:
657:
645:
643:
642:
637:
625:
623:
622:
617:
605:
603:
602:
597:
585:
583:
582:
577:
557:
555:
554:
549:
506:
504:
503:
498:
465:
463:
462:
457:
443:Chromatic number
437:
435:
434:
429:
427:
424:
420:
417:
321:
319:
318:
313:
311:
308:
304:
301:
235:
233:
232:
227:
225:
222:
218:
215:
201:
141:
139:
138:
133:
131:
126:
125:
116:
114:
110:
109:
101:
72:
70:
69:
64:
31:
19:
18:
2175:
2174:
2170:
2169:
2168:
2166:
2165:
2164:
2145:
2144:
2084:
2079:
2065:10.2307/2319046
2004:10.1007/b105810
1970:math.CO/0506260
1880:
1868:
1830:
1825:
1824:
1816:
1812:
1804:
1800:
1792:
1788:
1780:
1776:
1768:
1764:
1756:
1752:
1744:
1740:
1735:
1604:
1572:maximal cliques
1546:
1526:
1523:
1522:
1458:
1419:
1375:
1355:
1324:
1320:
1292:
1282:
1277:
1275:
1273:
1270:
1269:
1219:
1217:Turán's theorem
1213:
1211:Turán's theorem
1186:
1183:
1182:
1166:
1163:
1162:
1146:
1143:
1142:
1116:
1113:
1112:
1084:
1081:
1080:
1064:
1061:
1060:
1038:
1035:
1034:
1012:
1009:
1008:
992:
989:
988:
961:
957:
955:
940:
933:
929:
928:
924:
922:
919:
918:
896:
893:
892:
870:
857:
856:
855:
840:
836:
827:
823:
822:
820:
805:
798:
794:
792:
789:
788:
732:
728:
726:
723:
722:
691:
688:
687:
671:
668:
667:
651:
648:
647:
631:
628:
627:
611:
608:
607:
591:
588:
587:
571:
568:
567:
528:
525:
524:
477:
474:
473:
451:
448:
447:
422:
421:
416:
414:
408:
407:
396:
390:
389:
348:
341:
337:
335:
332:
331:
306:
305:
300:
298:
292:
291:
280:
274:
273:
262:
255:
251:
249:
246:
245:
220:
219:
214:
212:
206:
205:
197:
186:
180:
179:
168:
161:
157:
155:
152:
151:
121:
117:
115:
100:
93:
89:
87:
84:
83:
58:
55:
54:
34:
17:
12:
11:
5:
2173:
2163:
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2157:
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2142:
2123:
2104:
2083:
2082:External links
2080:
2078:
2077:
2046:
2030:
2022:Roberts, F. S.
2018:
1990:Wood, David R.
1985:
1963:(6): 774–780.
1946:
1911:
1893:(2): 139–153.
1873:
1861:
1849:(2): 139–143.
1831:
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1786:
1774:
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1758:Roberts (1969)
1750:
1737:
1736:
1734:
1731:
1685:-vertex graph
1664:grid embedding
1612:disjoint union
1603:
1600:
1578: + 2
1559:
1556:
1553:
1549:
1545:
1542:
1539:
1536:
1533:
1530:
1456:
1446:cross-polytope
1422:Roberts (1969)
1414:
1409:complete graph
1373:
1367:cross polytope
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1323:
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1215:Main article:
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586:vertices into
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2131:"Turán Graph"
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1988:Pór, Attila;
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1914:Moon, J. W.;
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1444:-dimensional
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1434:Roberts graph
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1353:Special cases
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1139:regular graph
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2013:11693/27422
1639:eigenvalues
1519:few cliques
1496:Moore graph
1494:is even, a
1181:(i.e. when
521:Turán graph
39:Named after
22:Turán graph
2149:Categories
2044:: 436–452.
2028:: 301–310.
1828:References
1709:admits an
1616:complement
1490:and, when
1465:octahedron
1363:octahedron
1137:. It is a
2136:MathWorld
2117:MathWorld
2098:MathWorld
2034:Turán, P.
1928:: 23–28.
1916:Moser, L.
1574:, where 3
1555:⌉
1541:⌈
1508:symmetric
1486:,2) is a
1312:−
1309:α
1284:−
1223:Pál Turán
1122:−
1096:−
1090:−
1044:−
938:−
902:≤
834:−
803:−
758:…
418:otherwise
381:≤
375:∧
369:≤
360:∨
346:∞
302:otherwise
260:∞
216:otherwise
192:≤
166:∞
98:−
43:Pál Turán
1907:17854032
1691:subgraph
1438:skeleton
1428:exactly
1426:boxicity
974:⌋
926:⌊
470:Notation
241:Diameter
50:Vertices
2073:2319046
1942:9855414
1608:cograph
1570:has 32
1498:. When
1459:is the
1407:from a
558:, is a
2071:
1940:
1905:
1440:of an
1365:, a 3-
1238:clique
1033:, and
147:Radius
2069:JSTOR
1965:arXiv
1938:S2CID
1903:S2CID
1881:(PDF)
1869:(PDF)
1733:Notes
1713:with
1689:is a
1467:. If
1457:2,2,2
1420:. As
1374:2,2,2
327:Girth
78:Edges
1614:and
1586:and
1510:and
1361:The
891:For
686:(so
626:and
519:The
2061:doi
2008:hdl
2000:doi
1975:doi
1961:307
1930:doi
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1953:.
1936:.
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1891:12
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1660:rn
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2010::
2002::
1983:.
1977::
1967::
1944:.
1932::
1926:3
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