120:
265:
In conferences held at
Oberwolfach, it is the custom for the participants to dine together in a room with circular tables, not all the same size, and with assigned seating that rearranges the participants from meal to meal. The Oberwolfach problem asks how to make a seating chart for a given set of
587:
In order for a solution to exist, the total number of conference participants (or equivalently, the total capacity of the tables, or the total number of vertices of the given cycle graphs) must be an odd number. For, at each meal, each participant sits next to two neighbors, so the total number of
460:
1172:, of grouping fifteen schoolgirls into rows of three in seven different ways so that each pair of girls appears once in each triple, is a special case of the Oberwolfach problem,
588:
neighbors of each participant must be even, and this is only possible when the total number of participants is odd. The problem has, however, also been extended to even values of
266:
tables so that all tables are full at each meal and all pairs of conference participants are seated next to each other exactly once. An instance of the problem can be denoted as
314:
352:
188:
1209:
982:
943:
904:
854:
772:
733:
391:
1927:
1014:
2003:
1489:
1382:
1158:
1120:
1082:
810:
1891:
1429:
1272:
1240:
542:
220:
148:
88:
686:
250:
1449:
1402:
1342:
1322:
1298:
1037:
658:
626:
606:
582:
562:
515:
491:
108:
61:
41:
1641:
1280:, on the decomposition of a complete graph into cycles of given sizes, is related to the Oberwolfach problem, but neither is a special case of the other. If
860:
is supported by recent non-constructive and asymptotic solutions for large complete graphs of order greater than a lower bound that is however unquantified.
640:(a different mathematical problem involving seating arrangements of diners and tables), this variant of the problem can be formulated by supposing that the
2216:
113:
1761:
354:
are the given table sizes. Alternatively, when some table sizes are repeated, they may be denoted using exponential notation; for instance,
688:
married couples, and that the seating arrangements should place each diner next to each other diner except their own spouse exactly once.
396:
Formulated as a problem in graph theory, the pairs of people sitting next to each other at a single meal can be represented as a
2170:
2081:
1832:
1697:
1169:
2211:
1931:
1521:
2079:
Bryant, Darryn; Scharaschkin, Victor (2009), "Complete solutions to the
Oberwolfach problem for an infinite set of orders",
237:
that may be formulated either as a problem of scheduling seating assignments for diners, or more abstractly as a problem in
2124:
1971:
1324:
vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the
Oberwolfach problem for
1212:
406:
397:
2012:
1789:
1277:
269:
1618:
1451:, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have
462:
of the specified lengths, with one cycle for each of the dining tables. This union of cycles is a
319:
2133:
1706:
1650:
1608:
153:
1175:
948:
909:
870:
815:
738:
699:
357:
1896:
1757:
993:
637:
1975:
1454:
1347:
1125:
1087:
1049:
777:
2179:
2143:
2090:
2021:
1940:
1841:
1830:; Wagner, David (1989), "The Oberwolfach problem and factors of uniform odd length cycles",
1798:
1749:
1716:
1660:
1573:
1530:
629:
242:
2193:
2155:
2102:
2066:
2033:
1954:
1869:
1853:
1810:
1771:
1728:
1672:
1587:
1544:
1431:
into this many cycles of each size can be grouped into disjoint cycles that form copies of
1407:
1245:
1218:
520:
193:
126:
66:
2189:
2151:
2098:
2062:
2029:
1950:
1849:
1806:
1767:
1724:
1668:
1583:
1540:
1636:
663:
2120:"On factorisations of complete graphs into circulant graphs and the Oberwolfach problem"
2119:
1748:, North-Holland Math. Stud., vol. 115, Amsterdam: North-Holland, pp. 227–232,
1622:
1516:
1434:
1387:
1327:
1307:
1283:
1040:
1022:
643:
611:
591:
567:
547:
500:
476:
254:
246:
93:
46:
26:
2168:
Traetta, Tommaso (2013), "A complete solution to the two-table
Oberwolfach problems",
1753:
2205:
2115:
1945:
1845:
1823:
1784:
1685:
1578:
1535:
1512:
465:
696:
The only instances of the
Oberwolfach problem that are known not to be solvable are
2118:; Bryant, Darryn; Horsley, Daniel; Maenhaut, Barbara; Scharaschkin, Victor (2016),
1827:
1561:
238:
234:
2047:
401:
226:
2184:
2147:
2094:
1720:
1689:
857:
119:
1603:
Salassa, F.; Dragotto, G.; Traetta, T.; Buratti, M.; Della Croce, F. (2019),
1787:; Häggkvist, Roland (1985), "Some observations on the Oberwolfach problem",
1802:
1664:
856:. It is widely believed that all other instances have a solution. This
1564:; Rosa, Alexander (1979), "On a variation of the Oberwolfach problem",
15:
2025:
1605:
Merging
Combinatorial Design and Optimization: the Oberwolfach Problem
257:. It is known to be true for all sufficiently-large complete graphs.
1711:
1655:
1613:
2138:
118:
1639:; Osthus, Deryk (2021), "Resolution of the Oberwolfach problem",
2055:
1519:(1991), "A brief review on Egmont Köhler's mathematical work",
1866:
Hoffman, D. G.; Schellenberg, P. J. (1991), "The existence of
1344:
would also provide a decomposition of the complete graph into
628:, whether all of the edges of the complete graph except for a
1744:
Häggkvist, Roland (1985), "A lemma on cycle decompositions",
2046:
Deza, A.; Franek, F.; Hua, W.; Meszka, M.; Rosa, A. (2010),
2048:"Solutions to the Oberwolfach problem for orders 18 to 40"
987:
All instances in which all of the cycles have even length.
863:
Cases for which a constructive solution is known include:
564:
can be represented as an edge-disjoint union of copies of
393:
describes an instance with three tables of size five.
1978:
1899:
1872:
1457:
1437:
1410:
1390:
1350:
1330:
1310:
1286:
1248:
1221:
1178:
1128:
1090:
1052:
1025:
996:
990:
All instances (other than the known exceptions) with
951:
912:
873:
818:
780:
741:
702:
666:
646:
614:
594:
570:
550:
523:
517:
vertices, the question is whether the complete graph
503:
479:
409:
360:
322:
272:
196:
156:
129:
96:
69:
49:
29:
1997:
1921:
1885:
1483:
1443:
1423:
1396:
1376:
1336:
1316:
1292:
1266:
1234:
1203:
1152:
1114:
1076:
1031:
1008:
976:
937:
898:
848:
804:
766:
727:
680:
652:
620:
600:
576:
556:
536:
509:
485:
454:
385:
346:
308:
214:
182:
142:
102:
82:
55:
35:
190:, solving the Oberwolfach problem for the input
251:Oberwolfach Research Institute for Mathematics
8:
1642:Journal of the European Mathematical Society
90:be decomposed into edge-disjoint copies of
1635:Glock, Stefan; Joos, Felix; Kim, Jaehoon;
2183:
2137:
1983:
1977:
1944:
1904:
1898:
1877:
1871:
1710:
1654:
1612:
1577:
1534:
1473:
1456:
1436:
1415:
1409:
1389:
1366:
1349:
1329:
1309:
1285:
1247:
1226:
1220:
1192:
1177:
1127:
1089:
1051:
1024:
995:
965:
950:
926:
911:
887:
872:
817:
779:
755:
740:
716:
701:
670:
665:
645:
613:
593:
569:
549:
528:
522:
502:
478:
455:{\displaystyle C_{x}+C_{y}+C_{z}+\cdots }
440:
427:
414:
408:
374:
359:
321:
271:
253:, where the problem was posed in 1967 by
195:
174:
161:
155:
134:
128:
95:
74:
68:
48:
28:
1970:Bryant, Darryn; Danziger, Peter (2011),
1499:
1039:, belonging to infinite subsets of the
114:(more unsolved problems in mathematics)
1746:Cycles in graphs (Burnaby, B.C., 1982)
1598:
1596:
1404:. However, not every decomposition of
632:can be covered by copies of the given
1965:
1963:
1690:"The generalised Oberwolfach problem"
1555:
1553:
1019:All instances for certain choices of
7:
1507:
1505:
1503:
123:Decomposition of the complete graph
1739:
1737:
1972:"On bipartite 2-factorizations of
14:
2217:Unsolved problems in graph theory
1384:copies of each of the cycles of
1084:other than the known exceptions
309:{\displaystyle OP(x,y,z,\dots )}
2171:Journal of Combinatorial Theory
2082:Journal of Combinatorial Theory
1833:Journal of Combinatorial Theory
1698:Journal of Combinatorial Theory
18:Unsolved problem in mathematics
1470:
1458:
1363:
1351:
1261:
1255:
1198:
1185:
1147:
1135:
1109:
1097:
1071:
1059:
971:
958:
932:
919:
893:
880:
843:
825:
799:
787:
761:
748:
722:
709:
380:
367:
303:
279:
209:
197:
1:
2125:Ars Mathematica Contemporanea
1754:10.1016/S0304-0208(08)73015-9
2005:and the Oberwolfach problem"
1946:10.1016/0012-365X(91)90440-D
1846:10.1016/0097-3165(89)90059-9
1688:; Staden, Katherine (2022),
1579:10.1016/0012-365X(79)90162-6
1536:10.1016/0012-365X(91)90416-Y
1170:Kirkman's schoolgirl problem
347:{\displaystyle x,y,z,\dots }
183:{\displaystyle C_{3}+C_{4}}
2233:
2185:10.1016/j.jcta.2013.01.003
2148:10.26493/1855-3974.770.150
2095:10.1016/j.jctb.2009.03.003
1721:10.1016/j.jctb.2021.09.007
1242:is another special case,
1213:Hamiltonian decomposition
1204:{\displaystyle OP(3^{5})}
977:{\displaystyle OP(3^{4})}
938:{\displaystyle OP(3^{2})}
899:{\displaystyle OP(x^{y})}
849:{\displaystyle OP(3,3,5)}
767:{\displaystyle OP(3^{4})}
728:{\displaystyle OP(3^{2})}
660:diners are arranged into
386:{\displaystyle OP(5^{3})}
1922:{\displaystyle K_{2n}-F}
1009:{\displaystyle n\leq 60}
473:graph has this form. If
249:. It is named after the
2013:Journal of Graph Theory
1998:{\displaystyle K_{n}-I}
1826:; Schellenberg, P. J.;
1790:Journal of Graph Theory
1484:{\displaystyle (n-1)/2}
1377:{\displaystyle (n-1)/2}
1153:{\displaystyle OP(4,5)}
1115:{\displaystyle OP(3,3)}
1077:{\displaystyle OP(x,y)}
805:{\displaystyle OP(4,5)}
63:can the complete graph
1999:
1923:
1887:
1803:10.1002/jgt.3190090114
1491:copies of each cycle.
1485:
1445:
1425:
1398:
1378:
1338:
1318:
1294:
1268:
1236:
1205:
1154:
1116:
1078:
1033:
1010:
978:
939:
900:
850:
806:
768:
729:
682:
654:
622:
602:
578:
558:
538:
511:
487:
456:
387:
348:
310:
222:
216:
184:
144:
104:
84:
57:
37:
2212:Mathematical problems
2000:
1924:
1888:
1886:{\displaystyle C_{k}}
1486:
1446:
1426:
1424:{\displaystyle K_{n}}
1399:
1379:
1339:
1319:
1295:
1269:
1267:{\displaystyle OP(n)}
1237:
1235:{\displaystyle K_{n}}
1206:
1155:
1117:
1079:
1034:
1011:
979:
940:
901:
851:
807:
769:
730:
683:
655:
623:
608:by asking, for those
603:
579:
559:
539:
537:{\displaystyle K_{n}}
512:
488:
457:
388:
349:
311:
217:
215:{\displaystyle (3,4)}
185:
150:into three copies of
145:
143:{\displaystyle K_{7}}
122:
105:
85:
83:{\displaystyle K_{n}}
58:
38:
1976:
1932:Discrete Mathematics
1897:
1870:
1566:Discrete Mathematics
1522:Discrete Mathematics
1455:
1435:
1408:
1388:
1348:
1328:
1308:
1284:
1278:Alspach's conjecture
1246:
1219:
1215:of a complete graph
1176:
1126:
1088:
1050:
1023:
994:
949:
910:
871:
816:
778:
739:
700:
664:
644:
612:
592:
568:
548:
521:
501:
477:
407:
358:
320:
270:
194:
154:
127:
94:
67:
47:
27:
23:For which 2-regular
1893:-factorizations of
1623:2019arXiv190312112S
681:{\displaystyle n/2}
231:Oberwolfach problem
1995:
1919:
1883:
1560:Huang, Charlotte;
1481:
1441:
1421:
1394:
1374:
1334:
1314:
1290:
1264:
1232:
1201:
1150:
1112:
1074:
1029:
1006:
974:
935:
896:
846:
802:
764:
725:
678:
650:
618:
598:
574:
554:
534:
507:
483:
452:
383:
344:
306:
223:
212:
180:
140:
100:
80:
53:
33:
2026:10.1002/jgt.20538
1763:978-0-444-87803-8
1665:10.4171/jems/1060
1444:{\displaystyle G}
1397:{\displaystyle G}
1337:{\displaystyle G}
1317:{\displaystyle n}
1293:{\displaystyle G}
1211:. The problem of
1032:{\displaystyle n}
653:{\displaystyle n}
621:{\displaystyle n}
601:{\displaystyle n}
577:{\displaystyle G}
557:{\displaystyle n}
510:{\displaystyle n}
486:{\displaystyle G}
469:graph, and every
243:edge cycle covers
103:{\displaystyle G}
56:{\displaystyle G}
36:{\displaystyle n}
2224:
2197:
2196:
2187:
2165:
2159:
2158:
2141:
2112:
2106:
2105:
2076:
2070:
2069:
2052:
2043:
2037:
2036:
2009:
2004:
2002:
2001:
1996:
1988:
1987:
1967:
1958:
1957:
1948:
1939:(1–3): 243–250,
1928:
1926:
1925:
1920:
1912:
1911:
1892:
1890:
1889:
1884:
1882:
1881:
1863:
1857:
1856:
1820:
1814:
1813:
1781:
1775:
1774:
1741:
1732:
1731:
1714:
1694:
1682:
1676:
1675:
1658:
1649:(8): 2511–2547,
1632:
1626:
1625:
1616:
1600:
1591:
1590:
1581:
1557:
1548:
1547:
1538:
1509:
1490:
1488:
1487:
1482:
1477:
1450:
1448:
1447:
1442:
1430:
1428:
1427:
1422:
1420:
1419:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1370:
1343:
1341:
1340:
1335:
1323:
1321:
1320:
1315:
1303:
1299:
1297:
1296:
1291:
1273:
1271:
1270:
1265:
1241:
1239:
1238:
1233:
1231:
1230:
1210:
1208:
1207:
1202:
1197:
1196:
1165:Related problems
1159:
1157:
1156:
1151:
1121:
1119:
1118:
1113:
1083:
1081:
1080:
1075:
1038:
1036:
1035:
1030:
1015:
1013:
1012:
1007:
983:
981:
980:
975:
970:
969:
944:
942:
941:
936:
931:
930:
905:
903:
902:
897:
892:
891:
855:
853:
852:
847:
811:
809:
808:
803:
773:
771:
770:
765:
760:
759:
734:
732:
731:
726:
721:
720:
687:
685:
684:
679:
674:
659:
657:
656:
651:
636:graph. Like the
635:
630:perfect matching
627:
625:
624:
619:
607:
605:
604:
599:
583:
581:
580:
575:
563:
561:
560:
555:
543:
541:
540:
535:
533:
532:
516:
514:
513:
508:
496:
492:
490:
489:
484:
472:
468:
461:
459:
458:
453:
445:
444:
432:
431:
419:
418:
392:
390:
389:
384:
379:
378:
353:
351:
350:
345:
315:
313:
312:
307:
221:
219:
218:
213:
189:
187:
186:
181:
179:
178:
166:
165:
149:
147:
146:
141:
139:
138:
109:
107:
106:
101:
89:
87:
86:
81:
79:
78:
62:
60:
59:
54:
42:
40:
39:
34:
19:
2232:
2231:
2227:
2226:
2225:
2223:
2222:
2221:
2202:
2201:
2200:
2167:
2166:
2162:
2114:
2113:
2109:
2078:
2077:
2073:
2050:
2045:
2044:
2040:
2007:
1979:
1974:
1973:
1969:
1968:
1961:
1900:
1895:
1894:
1873:
1868:
1867:
1865:
1864:
1860:
1822:
1821:
1817:
1783:
1782:
1778:
1764:
1743:
1742:
1735:
1692:
1684:
1683:
1679:
1634:
1633:
1629:
1602:
1601:
1594:
1559:
1558:
1551:
1517:Ringel, Gerhard
1511:
1510:
1501:
1497:
1453:
1452:
1433:
1432:
1411:
1406:
1405:
1386:
1385:
1346:
1345:
1326:
1325:
1306:
1305:
1301:
1282:
1281:
1244:
1243:
1222:
1217:
1216:
1188:
1174:
1173:
1167:
1124:
1123:
1086:
1085:
1048:
1047:
1041:natural numbers
1021:
1020:
992:
991:
961:
947:
946:
922:
908:
907:
883:
869:
868:
814:
813:
776:
775:
751:
737:
736:
712:
698:
697:
694:
662:
661:
642:
641:
633:
610:
609:
590:
589:
566:
565:
546:
545:
524:
519:
518:
499:
498:
494:
475:
474:
470:
463:
436:
423:
410:
405:
404:
370:
356:
355:
318:
317:
268:
267:
263:
247:complete graphs
192:
191:
170:
157:
152:
151:
130:
125:
124:
117:
116:
111:
92:
91:
70:
65:
64:
45:
44:
43:-vertex graphs
25:
24:
21:
17:
12:
11:
5:
2230:
2228:
2220:
2219:
2214:
2204:
2203:
2199:
2198:
2178:(5): 984–997,
2160:
2132:(1): 157–173,
2116:Alspach, Brian
2107:
2089:(6): 904–918,
2071:
2038:
1994:
1991:
1986:
1982:
1959:
1918:
1915:
1910:
1907:
1903:
1880:
1876:
1858:
1828:Stinson, D. R.
1824:Alspach, Brian
1815:
1797:(1): 177–187,
1785:Alspach, Brian
1776:
1762:
1733:
1686:Keevash, Peter
1677:
1627:
1592:
1572:(3): 261–277,
1549:
1513:Lenz, Hanfried
1498:
1496:
1493:
1480:
1476:
1472:
1469:
1466:
1463:
1460:
1440:
1418:
1414:
1393:
1373:
1369:
1365:
1362:
1359:
1356:
1353:
1333:
1313:
1289:
1263:
1260:
1257:
1254:
1251:
1229:
1225:
1200:
1195:
1191:
1187:
1184:
1181:
1166:
1163:
1162:
1161:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1046:All instances
1044:
1028:
1017:
1005:
1002:
999:
988:
985:
973:
968:
964:
960:
957:
954:
934:
929:
925:
921:
918:
915:
895:
890:
886:
882:
879:
876:
867:All instances
845:
842:
839:
836:
833:
830:
827:
824:
821:
801:
798:
795:
792:
789:
786:
783:
763:
758:
754:
750:
747:
744:
724:
719:
715:
711:
708:
705:
693:
690:
677:
673:
669:
649:
638:ménage problem
617:
597:
573:
553:
531:
527:
506:
497:graph and has
482:
451:
448:
443:
439:
435:
430:
426:
422:
417:
413:
398:disjoint union
382:
377:
373:
369:
366:
363:
343:
340:
337:
334:
331:
328:
325:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
262:
259:
255:Gerhard Ringel
211:
208:
205:
202:
199:
177:
173:
169:
164:
160:
137:
133:
112:
99:
77:
73:
52:
32:
22:
16:
13:
10:
9:
6:
4:
3:
2:
2229:
2218:
2215:
2213:
2210:
2209:
2207:
2195:
2191:
2186:
2181:
2177:
2173:
2172:
2164:
2161:
2157:
2153:
2149:
2145:
2140:
2135:
2131:
2127:
2126:
2121:
2117:
2111:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2083:
2075:
2072:
2068:
2064:
2060:
2056:
2049:
2042:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2014:
2006:
1992:
1989:
1984:
1980:
1966:
1964:
1960:
1956:
1952:
1947:
1942:
1938:
1934:
1933:
1916:
1913:
1908:
1905:
1901:
1878:
1874:
1862:
1859:
1855:
1851:
1847:
1843:
1839:
1835:
1834:
1829:
1825:
1819:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1791:
1786:
1780:
1777:
1773:
1769:
1765:
1759:
1755:
1751:
1747:
1740:
1738:
1734:
1730:
1726:
1722:
1718:
1713:
1708:
1704:
1700:
1699:
1691:
1687:
1681:
1678:
1674:
1670:
1666:
1662:
1657:
1652:
1648:
1644:
1643:
1638:
1637:Kühn, Daniela
1631:
1628:
1624:
1620:
1615:
1610:
1606:
1599:
1597:
1593:
1589:
1585:
1580:
1575:
1571:
1567:
1563:
1562:Kotzig, Anton
1556:
1554:
1550:
1546:
1542:
1537:
1532:
1529:(1–3): 3–16,
1528:
1524:
1523:
1518:
1514:
1508:
1506:
1504:
1500:
1494:
1492:
1478:
1474:
1467:
1464:
1461:
1438:
1416:
1412:
1391:
1371:
1367:
1360:
1357:
1354:
1331:
1311:
1287:
1279:
1275:
1258:
1252:
1249:
1227:
1223:
1214:
1193:
1189:
1182:
1179:
1171:
1164:
1144:
1141:
1138:
1132:
1129:
1106:
1103:
1100:
1094:
1091:
1068:
1065:
1062:
1056:
1053:
1045:
1042:
1026:
1018:
1003:
1000:
997:
989:
986:
966:
962:
955:
952:
927:
923:
916:
913:
888:
884:
877:
874:
866:
865:
864:
861:
859:
840:
837:
834:
831:
828:
822:
819:
796:
793:
790:
784:
781:
756:
752:
745:
742:
717:
713:
706:
703:
692:Known results
691:
689:
675:
671:
667:
647:
639:
631:
615:
595:
585:
571:
551:
529:
525:
504:
480:
467:
449:
446:
441:
437:
433:
428:
424:
420:
415:
411:
403:
399:
394:
375:
371:
364:
361:
341:
338:
335:
332:
329:
326:
323:
300:
297:
294:
291:
288:
285:
282:
276:
273:
260:
258:
256:
252:
248:
244:
240:
236:
232:
228:
206:
203:
200:
175:
171:
167:
162:
158:
135:
131:
121:
115:
97:
75:
71:
50:
30:
2175:
2174:, Series A,
2169:
2163:
2129:
2123:
2110:
2086:
2085:, Series B,
2080:
2074:
2058:
2054:
2041:
2020:(1): 22–37,
2017:
2011:
1936:
1930:
1861:
1840:(1): 20–43,
1837:
1836:, Series A,
1831:
1818:
1794:
1788:
1779:
1745:
1702:
1701:, Series B,
1696:
1680:
1646:
1640:
1630:
1604:
1569:
1565:
1526:
1520:
1276:
1168:
862:
695:
586:
402:cycle graphs
395:
264:
239:graph theory
235:open problem
230:
224:
1705:: 281–318,
1304:graph with
261:Formulation
227:mathematics
2206:Categories
2061:: 95–102,
1712:2004.09937
1656:1806.04644
1614:1903.12112
1495:References
858:conjecture
2139:1411.6047
1990:−
1914:−
1465:−
1358:−
1302:2-regular
1001:≤
634:2-regular
544:of order
495:2-regular
471:2-regular
450:⋯
342:…
301:…
241:, on the
493:is this
2194:3033656
2156:3546656
2103:2558441
2067:2675892
2034:2833961
1955:1140806
1854:1008157
1811:0785659
1772:0821524
1729:4332743
1673:4269420
1619:Bibcode
1588:0541472
1545:1140782
906:except
466:regular
2192:
2154:
2101:
2065:
2032:
1953:
1852:
1809:
1770:
1760:
1727:
1671:
1586:
1543:
812:, and
316:where
233:is an
229:, the
2134:arXiv
2051:(PDF)
2008:(PDF)
1707:arXiv
1693:(PDF)
1651:arXiv
1609:arXiv
1300:is a
1758:ISBN
1122:and
945:and
2180:doi
2176:120
2144:doi
2091:doi
2022:doi
1941:doi
1929:",
1842:doi
1799:doi
1750:doi
1717:doi
1703:152
1661:doi
1574:doi
1531:doi
400:of
245:of
225:In
2208::
2190:MR
2188:,
2152:MR
2150:,
2142:,
2130:11
2128:,
2122:,
2099:MR
2097:,
2087:99
2063:MR
2059:74
2057:,
2053:,
2030:MR
2028:,
2018:68
2016:,
2010:,
1962:^
1951:MR
1949:,
1937:97
1935:,
1850:MR
1848:,
1838:52
1807:MR
1805:,
1793:,
1768:MR
1766:,
1756:,
1736:^
1725:MR
1723:,
1715:,
1695:,
1669:MR
1667:,
1659:,
1647:23
1645:,
1617:,
1607:,
1595:^
1584:MR
1582:,
1570:27
1568:,
1552:^
1541:MR
1539:,
1527:97
1525:,
1515:;
1502:^
1274:.
1004:60
774:,
735:,
584:.
464:2-
2182::
2146::
2136::
2093::
2024::
1993:I
1985:n
1981:K
1943::
1917:F
1909:n
1906:2
1902:K
1879:k
1875:C
1844::
1801::
1795:9
1752::
1719::
1709::
1663::
1653::
1621::
1611::
1576::
1533::
1479:2
1475:/
1471:)
1468:1
1462:n
1459:(
1439:G
1417:n
1413:K
1392:G
1372:2
1368:/
1364:)
1361:1
1355:n
1352:(
1332:G
1312:n
1288:G
1262:)
1259:n
1256:(
1253:P
1250:O
1228:n
1224:K
1199:)
1194:5
1190:3
1186:(
1183:P
1180:O
1160:.
1148:)
1145:5
1142:,
1139:4
1136:(
1133:P
1130:O
1110:)
1107:3
1104:,
1101:3
1098:(
1095:P
1092:O
1072:)
1069:y
1066:,
1063:x
1060:(
1057:P
1054:O
1043:.
1027:n
1016:.
998:n
984:.
972:)
967:4
963:3
959:(
956:P
953:O
933:)
928:2
924:3
920:(
917:P
914:O
894:)
889:y
885:x
881:(
878:P
875:O
844:)
841:5
838:,
835:3
832:,
829:3
826:(
823:P
820:O
800:)
797:5
794:,
791:4
788:(
785:P
782:O
762:)
757:4
753:3
749:(
746:P
743:O
723:)
718:2
714:3
710:(
707:P
704:O
676:2
672:/
668:n
648:n
616:n
596:n
572:G
552:n
530:n
526:K
505:n
481:G
447:+
442:z
438:C
434:+
429:y
425:C
421:+
416:x
412:C
381:)
376:3
372:5
368:(
365:P
362:O
339:,
336:z
333:,
330:y
327:,
324:x
304:)
298:,
295:z
292:,
289:y
286:,
283:x
280:(
277:P
274:O
210:)
207:4
204:,
201:3
198:(
176:4
172:C
168:+
163:3
159:C
136:7
132:K
110:?
98:G
76:n
72:K
51:G
31:n
20::
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