1751:
1147:
425:
29:
946:
752:
1137:
with only two colors. The two colors of this coloring may be found from the subset construction of hypercube graphs, by giving one color to the subsets that have an even number of elements and the other color to the subsets with an odd number of elements.
1570:
1478:
223:
472:
elements, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single element. Equivalently, it may be constructed using
626:
1227:
A lesser known fact is that every perfect matching in the hypercube extends to a
Hamiltonian cycle. The question whether every matching extends to a Hamiltonian cycle remains an open problem.
941:{\displaystyle A_{n}={\begin{cases}1_{2}\otimes _{K}A_{n-1}+A_{1}\otimes _{K}1_{2^{n-1}}&{\text{if }}n>1\\{\begin{bmatrix}0&1\\1&0\end{bmatrix}}&{\text{if }}n=1\end{cases}}}
743:
1611:
1697:
1111:
655:
681:
488:
of their labels is one. These two constructions are closely related: a binary number may be interpreted as a set (the set of positions where it has a
1802:
connecting each hypercube vertex to another vertex with which it should be connected, there is always a way to connect these pairs of vertices by
1493:
1389:
1196:
on the dimension of the hypercube, and the construction of the hypercube graph by joining two smaller hypercubes with a matching.
2182:
492:
digit), and two such sets differ in a single element whenever the corresponding two binary numbers have
Hamming distance one.
139:
1879:
268:
576:
357:-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each
1966:
2019:
688:
2070:
1845:
1262:
963:
746:. The sum of these two terms gives a recursive function function for the adjacency matrix of a hypercube:
370:
1582:
1373:
1791:
1663:
1283:
perfect matchings. (this is another consequence that follows easily from the inductive construction.)
774:
1340:
365:, with two vertices adjacent when their binary representations differ in a single digit. It is the
1835:
1739:
2187:
1320:
in the
Euclidean plane by using the construction of the hypercube graph from subsets of a set of
400:, which are graphs that have exactly three edges touching each vertex. The only hypercube graph
74:
1077:
534:
to the corresponding vertex in the other copy, as shown in the figure. The joining edges form a
1193:
1090:
236:
1820:
1347:
321:
132:
47:
2119:
Szymanski, Ted H. (1989), "On the
Permutation Capability of a Circuit-Switched Hypercube",
2056:
1999:
1287:
1025:
633:
86:
8:
1803:
1317:
240:
59:
1916:
1150:
A Hamiltonian cycle on a tesseract with vertices labelled with a 4-bit cyclic Gray code
666:
105:
2078:
2155:
2038:
1875:
1840:
1830:
1576:
1081:
1053:
557:
465:
244:
2159:
2151:
2099:
2074:
2042:
2034:
1943:
1908:
1795:
1785:
1781:
1617:
1177:
1173:
542:
535:
485:
390:
120:
2052:
1995:
1934:
Fink, J. (2007), "Perfect matchings extend to
Hamiltonian cycles in hypercubes",
1815:
1487:
1306:
1291:
1130:
1057:
979:. More generally the Cartesian product of copies of a complete graph is called a
659:
248:
232:
2142:; Hayes, J. P.; Wu, H.-J. (1988), "A survey of the theory of hypercube graphs",
310:
is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube.
2123:, vol. 1, Silver Spring, MD: IEEE Computer Society Press, pp. 103–110
1979:
1948:
1825:
1703:
1295:
1134:
1010:
507:
374:
1750:
2176:
1480:
1358:
1313:
1258:
980:
481:
362:
332:
299:
1192:-coloring of the graph. Both facts are easy to prove using the principle of
2139:
2104:
2015:
1963:
1850:
1777:
1354:
1273:
1269:
277:
1211:-bit cyclic Gray codes and the set of Hamiltonian cycles in the hypercube
2164:
2047:
1799:
1325:
397:
541:
The above construction gives a recursive algorithm for constructing the
424:
1920:
1200:
1113:
1073:
1045:
2069:
Optimal
Numberings and Isoperimetric Problems on Graphs, L.H. Harper,
1328:
for each set element, and placing the vertex corresponding to the set
1146:
1204:
295:
1912:
1199:
Hamiltonicity of the hypercube is tightly related to the theory of
1565:{\displaystyle \sum _{i=0}^{n-1}{\binom {i}{\lfloor i/2\rfloor }}}
684:
dimensions. Meanwhile the joining edges have an adjacency matrix
461:
1798:
for communications. It states that, no matter how one chooses a
1176:, a cycle that visits each vertex exactly once. Additionally, a
28:
2090:
Roichman, Y. (2000), "On the
Achromatic Number of Hypercubes",
437:
by connecting pairs of corresponding vertices in two copies of
350:
1613:, but the constant of proportionality is not known precisely.
1085:
1473:{\displaystyle 2^{2^{n}-n-1}\prod _{k=2}^{n}k^{n \choose k}}
1294:. The symmetries of hypercube graphs can be represented as
1648:
and as the eigenvalues of its
Laplacian matrix the numbers
1049:
934:
1872:
Across the Board: The
Mathematics of Chessboard Problems
218:{\displaystyle \{(n-2k)^{\binom {n}{k}};k=0,\ldots ,n\}}
983:; the hypercube graphs are examples of Hamming graphs.
884:
349:
may also be constructed by creating a vertex for each
290:
is the graph formed from the vertices and edges of an
1666:
1585:
1496:
1392:
1093:
755:
691:
669:
636:
579:
142:
1967:
522:, by adding an edge from each vertex in one copy of
484:
and connecting two vertices by an edge whenever the
1276:, and can be formed as a retraction of a hypercube.
621:{\displaystyle \mathrm {1} _{2}\otimes _{K}A_{n-1}}
1895:Mills, W. H. (1963), "Some complete cycles on the
1768:) in the snakes-in-the-box problem for dimensions
1691:
1605:
1564:
1472:
1105:
940:
737:
675:
649:
620:
217:
1683:
1670:
1556:
1527:
16:Graphs formed by a hypercube's edges and vertices
2174:
1901:Proceedings of the American Mathematical Society
1982:(1955), "Über drei kombinatorische Probleme am
1188:if and only if they have different colors in a
2144:Computers & Mathematics with Applications
2138:
2027:Computers & Mathematics with Applications
2014:
1907:(4), American Mathematical Society: 640–643,
1794:concerns the suitability of a hypercube as a
1463:
1450:
396:Hypercube graphs should not be confused with
181:
168:
2121:Proc. Internat. Conf. on Parallel Processing
2020:"A survey of the theory of hypercube graphs"
1550:
1536:
738:{\displaystyle A_{1}\otimes _{K}1_{2^{n-1}}}
212:
143:
1986:-dimensionalen Wiirfel und Wiirfelgitter",
1784:of a given hypercube graph is known as the
407:that is a cubic graph is the cubical graph
377:, and may be decomposed into two copies of
1874:, Princeton University Press, p. 68,
1220:. An analogous property holds for acyclic
2163:
2118:
2103:
2092:Journal of Combinatorial Theory, Series B
2046:
1947:
1936:Journal of Combinatorial Theory, Series B
2089:
2018:; Hayes, John P.; Wu, Horng-Jyh (1988),
1749:
1145:
423:
1869:
1224:-bit Gray codes and Hamiltonian paths.
2175:
1978:
460:may be constructed from the family of
2010:
2008:
1972:
1894:
1806:that do not share any directed edge.
1933:
1230:
1000:consists of a single vertex, while
13:
2005:
1674:
1531:
1454:
1361:with no crossings) if and only if
1301:contains all the cycles of length
1207:correspondence between the set of
172:
14:
2199:
1274:isometric subgraph of a hypercube
1052:and is a planar graph with eight
1606:{\displaystyle {\sqrt {n2^{n}}}}
1141:
1124:
27:
2071:Journal of Combinatorial Theory
1846:Hypercube internetwork topology
1692:{\displaystyle {\binom {n}{k}}}
1660:th eigenvalue has multiplicity
419:
2112:
2083:
2063:
1956:
1927:
1888:
1863:
1616:has as the eigenvalues of its
1324:elements, choosing a distinct
269:Table of graphs and parameters
162:
146:
1:
2183:Parametric families of graphs
2132:
2079:10.1016/S0021-9800(66)80059-5
1988:Abh. Math. Sem. Univ. Hamburg
1969:on Open Problem Garden. 2007.
1332:at the sum of the vectors in
1119:
389:connected to each other by a
2156:10.1016/0898-1221(88)90213-1
2039:10.1016/0898-1221(88)90213-1
1203:. More precisely there is a
1180:exists between two vertices
560:, so that the two copies of
506:may be constructed from the
339:edges touching each vertex.
7:
1809:
1776:The problem of finding the
1754:Maximum lengths of snakes (
1745:
1272:. Every median graph is an
986:
970:two-vertex complete graphs
10:
2204:
1949:10.1016/j.jctb.2007.02.007
556:. Copying is done via the
1870:Watkins, John J. (2004),
1129:Every hypercube graph is
1106:{\displaystyle 4\times 4}
572:have an adjacency matrix
267:
254:
228:
131:
119:
104:
85:
73:
58:
46:
26:
21:
1856:
951:Another construction of
1368:. For larger values of
1344:-vertex-connected graph
2105:10.1006/jctb.2000.1955
1792:Szymanski's conjecture
1773:
1693:
1607:
1566:
1523:
1474:
1442:
1151:
1107:
949:
942:
739:
677:
651:
622:
476:vertices labeled with
446:
219:
1821:Cube-connected cycles
1753:
1740:Lévy family of graphs
1694:
1608:
1567:
1497:
1475:
1422:
1149:
1108:
943:
748:
740:
678:
652:
650:{\displaystyle 1_{d}}
623:
427:
220:
2073:, 1, 385–393,
1780:or cycle that is an
1704:isoperimetric number
1664:
1583:
1494:
1390:
1372:, the hypercube has
1235:The hypercube graph
1091:
1078:Möbius configuration
753:
689:
667:
634:
577:
449:The hypercube graph
342:The hypercube graph
298:. For instance, the
140:
33:The hypercube graph
1318:unit distance graph
1296:signed permutations
1774:
1689:
1603:
1562:
1470:
1348:Balinski's theorem
1152:
1103:
938:
933:
909:
735:
673:
647:
618:
510:of two hypercubes
447:
373:of the two-vertex
215:
1881:978-0-691-15498-5
1841:Halved cube graph
1836:Frankl–Rödl graph
1831:Folded cube graph
1681:
1601:
1577:achromatic number
1554:
1461:
1307:bipancyclic graph
1174:Hamiltonian cycle
1170: > 1
1080:. It is also the
1013:on two vertices.
964:Cartesian product
920:
864:
676:{\displaystyle d}
558:Kronecker product
371:Cartesian product
274:
273:
179:
2195:
2168:
2167:
2126:
2124:
2116:
2110:
2108:
2107:
2087:
2081:
2067:
2061:
2059:
2050:
2024:
2012:
2003:
2002:
1985:
1976:
1970:
1960:
1954:
1952:
1951:
1942:(6): 1074–1076,
1931:
1925:
1923:
1898:
1892:
1886:
1884:
1867:
1796:network topology
1786:snake-in-the-box
1782:induced subgraph
1737:
1730:
1715:
1698:
1696:
1695:
1690:
1688:
1687:
1686:
1673:
1659:
1655:
1647:
1618:adjacency matrix
1612:
1610:
1609:
1604:
1602:
1600:
1599:
1587:
1579:proportional to
1571:
1569:
1568:
1563:
1561:
1560:
1559:
1553:
1546:
1530:
1522:
1511:
1479:
1477:
1476:
1471:
1469:
1468:
1467:
1466:
1453:
1441:
1436:
1421:
1420:
1407:
1406:
1382:
1371:
1367:
1343:
1335:
1331:
1323:
1304:
1282:
1252:
1245:
1231:Other properties
1219:
1210:
1191:
1187:
1183:
1178:Hamiltonian path
1171:
1164:
1154:Every hypercube
1112:
1110:
1109:
1104:
1071:
1043:
1031:
1023:
1008:
999:
978:
969:
961:
947:
945:
944:
939:
937:
936:
921:
918:
914:
913:
865:
862:
858:
857:
856:
855:
835:
834:
825:
824:
812:
811:
796:
795:
786:
785:
765:
764:
745:
744:
742:
741:
736:
734:
733:
732:
731:
711:
710:
701:
700:
683:
682:
680:
679:
674:
657:
656:
654:
653:
648:
646:
645:
628:
627:
625:
624:
619:
617:
616:
601:
600:
591:
590:
585:
571:
555:
545:of a hypercube,
543:adjacency matrix
536:perfect matching
533:
521:
505:
491:
486:Hamming distance
479:
475:
471:
459:
445:
436:
428:Construction of
415:
406:
391:perfect matching
388:
368:
360:
356:
348:
338:
331:edges, and is a
330:
320:
316:
309:
293:
289:
263:
237:Distance regular
224:
222:
221:
216:
187:
186:
185:
184:
171:
127:
121:Chromatic number
115:
100:
93:
81:
69:
54:
41:
31:
19:
18:
2203:
2202:
2198:
2197:
2196:
2194:
2193:
2192:
2173:
2172:
2135:
2130:
2129:
2117:
2113:
2088:
2084:
2068:
2064:
2022:
2013:
2006:
1983:
1977:
1973:
1962:Ruskey, F. and
1961:
1957:
1932:
1928:
1913:10.2307/2034292
1896:
1893:
1889:
1882:
1868:
1864:
1859:
1816:de Bruijn graph
1812:
1767:
1760:
1748:
1732:
1729:
1721:
1706:
1682:
1669:
1668:
1667:
1665:
1662:
1661:
1657:
1649:
1621:
1595:
1591:
1586:
1584:
1581:
1580:
1555:
1542:
1535:
1526:
1525:
1524:
1512:
1501:
1495:
1492:
1491:
1462:
1449:
1448:
1447:
1443:
1437:
1426:
1402:
1398:
1397:
1393:
1391:
1388:
1387:
1381:− 4)2 + 1
1376:
1369:
1362:
1341:
1333:
1329:
1321:
1302:
1280:
1263:Boolean algebra
1247:
1244:
1236:
1233:
1218:
1212:
1208:
1189:
1185:
1181:
1166:
1163:
1155:
1144:
1127:
1122:
1092:
1089:
1088:
1070:
1064:
1042:
1036:
1029:
1028:of length
1022:
1016:
1007:
1001:
998:
992:
989:
977:
971:
967:
960:
952:
932:
931:
917:
915:
908:
907:
902:
896:
895:
890:
880:
879:
876:
875:
861:
859:
845:
841:
840:
836:
830:
826:
820:
816:
801:
797:
791:
787:
781:
777:
770:
769:
760:
756:
754:
751:
750:
721:
717:
716:
712:
706:
702:
696:
692:
690:
687:
686:
685:
668:
665:
664:
663:
660:identity matrix
641:
637:
635:
632:
631:
630:
606:
602:
596:
592:
586:
581:
580:
578:
575:
574:
573:
570:
561:
554:
546:
532:
523:
520:
511:
504:
496:
495:Alternatively,
489:
477:
473:
469:
458:
450:
444:
438:
435:
429:
422:
414:
408:
405:
401:
387:
378:
366:
358:
354:
347:
343:
336:
325:
318:
315:
311:
308:
302:
291:
288:
284:
282:hypercube graph
262:
258:
247:
243:
239:
235:
180:
167:
166:
165:
161:
141:
138:
137:
125:
110:
95:
91:
79:
64:
52:
42:
40:
34:
22:Hypercube graph
17:
12:
11:
5:
2201:
2191:
2190:
2188:Regular graphs
2185:
2171:
2170:
2150:(4): 277–289,
2134:
2131:
2128:
2127:
2111:
2098:(2): 177–182,
2082:
2062:
2033:(4): 277–289,
2004:
1971:
1955:
1926:
1887:
1880:
1861:
1860:
1858:
1855:
1854:
1853:
1848:
1843:
1838:
1833:
1828:
1826:Fibonacci cube
1823:
1818:
1811:
1808:
1765:
1758:
1747:
1744:
1725:
1718:
1717:
1700:
1699:in both cases.
1685:
1680:
1677:
1672:
1614:
1598:
1594:
1590:
1573:
1558:
1552:
1549:
1545:
1541:
1538:
1534:
1529:
1521:
1518:
1515:
1510:
1507:
1504:
1500:
1484:
1481:spanning trees
1465:
1460:
1457:
1452:
1446:
1440:
1435:
1432:
1429:
1425:
1419:
1416:
1413:
1410:
1405:
1401:
1396:
1384:
1351:
1337:
1310:
1305:and is thus a
1299:
1288:arc transitive
1284:
1279:has more than
1277:
1266:
1240:
1232:
1229:
1216:
1159:
1143:
1140:
1126:
1123:
1121:
1118:
1102:
1099:
1096:
1082:knight's graph
1068:
1040:
1020:
1011:complete graph
1005:
996:
988:
985:
975:
956:
935:
930:
927:
924:
916:
912:
906:
903:
901:
898:
897:
894:
891:
889:
886:
885:
883:
878:
877:
874:
871:
868:
860:
854:
851:
848:
844:
839:
833:
829:
823:
819:
815:
810:
807:
804:
800:
794:
790:
784:
780:
776:
775:
773:
768:
763:
759:
730:
727:
724:
720:
715:
709:
705:
699:
695:
672:
644:
640:
615:
612:
609:
605:
599:
595:
589:
584:
565:
550:
527:
515:
508:disjoint union
500:
482:binary numbers
454:
442:
433:
421:
418:
412:
403:
382:
375:complete graph
345:
313:
306:
286:
272:
271:
265:
264:
260:
256:
252:
251:
230:
226:
225:
214:
211:
208:
205:
202:
199:
196:
193:
190:
183:
178:
175:
170:
164:
160:
157:
154:
151:
148:
145:
135:
129:
128:
123:
117:
116:
108:
102:
101:
89:
83:
82:
77:
71:
70:
62:
56:
55:
50:
44:
43:
38:
32:
24:
23:
15:
9:
6:
4:
3:
2:
2200:
2189:
2186:
2184:
2181:
2180:
2178:
2166:
2165:2027.42/27522
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2136:
2122:
2115:
2106:
2101:
2097:
2093:
2086:
2080:
2076:
2072:
2066:
2058:
2054:
2049:
2048:2027.42/27522
2044:
2040:
2036:
2032:
2028:
2021:
2017:
2016:Harary, Frank
2011:
2009:
2001:
1997:
1993:
1989:
1981:
1975:
1968:
1965:
1959:
1950:
1945:
1941:
1937:
1930:
1922:
1918:
1914:
1910:
1906:
1902:
1891:
1883:
1877:
1873:
1866:
1862:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1813:
1807:
1805:
1801:
1797:
1793:
1789:
1787:
1783:
1779:
1771:
1764:
1761:) and coils (
1757:
1752:
1743:
1741:
1735:
1728:
1724:
1713:
1709:
1705:
1701:
1678:
1675:
1653:
1650:(0, 2, ..., 2
1645:
1641:
1637:
1633:
1629:
1625:
1619:
1615:
1596:
1592:
1588:
1578:
1574:
1547:
1543:
1539:
1532:
1519:
1516:
1513:
1508:
1505:
1502:
1498:
1489:
1485:
1482:
1458:
1455:
1444:
1438:
1433:
1430:
1427:
1423:
1417:
1414:
1411:
1408:
1403:
1399:
1394:
1385:
1380:
1375:
1365:
1360:
1356:
1352:
1349:
1345:
1338:
1327:
1319:
1315:
1311:
1308:
1300:
1297:
1293:
1289:
1285:
1278:
1275:
1271:
1267:
1264:
1260:
1259:Hasse diagram
1256:
1255:
1254:
1250:
1243:
1239:
1228:
1225:
1223:
1215:
1206:
1202:
1197:
1195:
1179:
1175:
1169:
1162:
1158:
1148:
1142:Hamiltonicity
1139:
1136:
1132:
1125:Bipartiteness
1117:
1115:
1100:
1097:
1094:
1087:
1083:
1079:
1075:
1067:
1061:
1059:
1055:
1051:
1047:
1039:
1033:
1027:
1019:
1014:
1012:
1004:
995:
984:
982:
981:Hamming graph
974:
965:
959:
955:
948:
928:
925:
922:
910:
904:
899:
892:
887:
881:
872:
869:
866:
852:
849:
846:
842:
837:
831:
827:
821:
817:
813:
808:
805:
802:
798:
792:
788:
782:
778:
771:
766:
761:
757:
747:
728:
725:
722:
718:
713:
707:
703:
697:
693:
670:
661:
642:
638:
613:
610:
607:
603:
597:
593:
587:
582:
568:
564:
559:
553:
549:
544:
539:
537:
530:
526:
518:
514:
509:
503:
499:
493:
487:
483:
467:
463:
457:
453:
441:
432:
426:
417:
411:
399:
394:
392:
385:
381:
376:
372:
364:
363:binary number
352:
340:
334:
333:regular graph
329:
323:
305:
301:
297:
294:-dimensional
283:
279:
270:
266:
257:
253:
250:
246:
242:
241:Unit distance
238:
234:
231:
227:
209:
206:
203:
200:
197:
194:
191:
188:
176:
173:
158:
155:
152:
149:
136:
134:
130:
124:
122:
118:
113:
109:
107:
106:Automorphisms
103:
98:
90:
88:
84:
78:
76:
72:
68:
63:
61:
57:
51:
49:
45:
37:
30:
25:
20:
2147:
2143:
2120:
2114:
2095:
2091:
2085:
2065:
2030:
2026:
1991:
1987:
1974:
1958:
1939:
1935:
1929:
1904:
1900:
1890:
1871:
1865:
1851:Partial cube
1790:
1778:longest path
1775:
1769:
1762:
1755:
1733:
1726:
1722:
1719:
1711:
1707:
1651:
1643:
1639:
1635:
1631:
1630:+ 2, −
1627:
1623:
1620:the numbers
1386:has exactly
1378:
1363:
1303:4, 6, ..., 2
1270:median graph
1261:of a finite
1248:
1241:
1237:
1234:
1226:
1221:
1213:
1198:
1167:
1160:
1156:
1153:
1133:: it can be
1128:
1065:
1062:
1037:
1034:
1017:
1015:
1002:
993:
990:
972:
957:
953:
950:
749:
566:
562:
551:
547:
540:
528:
524:
516:
512:
501:
497:
494:
455:
451:
448:
439:
430:
420:Construction
409:
398:cubic graphs
395:
383:
379:
341:
327:
303:
281:
278:graph theory
275:
111:
96:
66:
35:
1800:permutation
1772:from 1 to 4
1720:The family
1642:− 2,
1638:− 4,
1634:+ 4, ... ,
1326:unit vector
1056:and twelve
245:Hamiltonian
2177:Categories
2140:Harary, F.
2133:References
1980:Ringel, G.
1964:Savage, C.
1201:Gray codes
1120:Properties
1114:chessboard
1074:Levi graph
1063:The graph
1046:1-skeleton
1035:The graph
991:The graph
300:cube graph
229:Properties
1994:: 10–19,
1788:problem.
1626:, −
1551:⌋
1537:⌊
1517:−
1499:∑
1488:bandwidth
1424:∏
1415:−
1409:−
1292:symmetric
1253:) :
1205:bijective
1194:induction
1131:bipartite
1098:×
850:−
828:⊗
806:−
789:⊗
726:−
704:⊗
611:−
594:⊗
569:− 1
531:− 1
519:− 1
296:hypercube
249:Bipartite
233:Symmetric
204:…
153:−
1899:-cube",
1810:See also
1746:Problems
1731:for all
1622:(−
1490:exactly
1357:(can be
1086:toroidal
1054:vertices
987:Examples
919:if
863:if
322:vertices
255:Notation
133:Spectrum
75:Diameter
48:Vertices
2057:0949280
2000:0949280
1921:2034292
1312:can be
1257:is the
1135:colored
1076:of the
1072:is the
1044:is the
1009:is the
962:is the
658:is the
629:,where
462:subsets
361:-digit
280:, the
2055:
1998:
1919:
1878:
1736:> 1
1656:. The
1355:planar
1251:> 1
1172:has a
1084:for a
369:-fold
353:of an
351:subset
2023:(PDF)
1917:JSTOR
1857:Notes
1804:paths
1738:is a
1714:) = 1
1374:genus
1359:drawn
1346:, by
1339:is a
1316:as a
1314:drawn
1268:is a
1246:(for
1165:with
1058:edges
1048:of a
1026:cycle
1024:is a
480:-bit
468:with
464:of a
335:with
87:Girth
60:Edges
1876:ISBN
1702:has
1575:has
1486:has
1290:and
1184:and
1050:cube
870:>
317:has
2160:hdl
2152:doi
2100:doi
2075:doi
2043:hdl
2035:doi
1944:doi
1909:doi
1366:≤ 3
1353:is
1286:is
966:of
662:in
466:set
386:– 1
276:In
114:! 2
99:≥ 2
94:if
2179::
2158:,
2148:15
2146:,
2096:79
2094:,
2053:MR
2051:,
2041:,
2031:15
2029:,
2025:,
2007:^
1996:MR
1992:20
1990:,
1940:97
1938:,
1915:,
1905:14
1903:,
1742:.
1116:.
1060:.
1032:.
538:.
416:.
393:.
324:,
2169:.
2162::
2154::
2125:.
2109:.
2102::
2077::
2060:.
2045::
2037::
1984:n
1953:.
1946::
1924:.
1911::
1897:n
1885:.
1770:n
1766:c
1763:L
1759:s
1756:L
1734:n
1727:n
1723:Q
1716:.
1712:G
1710:(
1708:h
1684:)
1679:k
1676:n
1671:(
1658:k
1654:)
1652:n
1646:)
1644:n
1640:n
1636:n
1632:n
1628:n
1624:n
1597:n
1593:2
1589:n
1572:.
1557:)
1548:2
1544:/
1540:i
1533:i
1528:(
1520:1
1514:n
1509:0
1506:=
1503:i
1483:.
1464:)
1459:k
1456:n
1451:(
1445:k
1439:n
1434:2
1431:=
1428:k
1418:1
1412:n
1404:n
1400:2
1395:2
1383:.
1379:n
1377:(
1370:n
1364:n
1350:.
1342:n
1336:.
1334:S
1330:S
1322:n
1309:.
1298:.
1281:2
1265:.
1249:n
1242:n
1238:Q
1222:n
1217:n
1214:Q
1209:n
1190:2
1186:v
1182:u
1168:n
1161:n
1157:Q
1101:4
1095:4
1069:4
1066:Q
1041:3
1038:Q
1030:4
1021:2
1018:Q
1006:1
1003:Q
997:0
994:Q
976:2
973:K
968:n
958:n
954:Q
929:1
926:=
923:n
911:]
905:0
900:1
893:1
888:0
882:[
873:1
867:n
853:1
847:n
843:2
838:1
832:K
822:1
818:A
814:+
809:1
803:n
799:A
793:K
783:2
779:1
772:{
767:=
762:n
758:A
729:1
723:n
719:2
714:1
708:K
698:1
694:A
671:d
643:d
639:1
614:1
608:n
604:A
598:K
588:2
583:1
567:n
563:Q
552:n
548:A
529:n
525:Q
517:n
513:Q
502:n
498:Q
490:1
478:n
474:2
470:n
456:n
452:Q
443:2
440:Q
434:3
431:Q
413:3
410:Q
404:n
402:Q
384:n
380:Q
367:n
359:n
355:n
346:n
344:Q
337:n
328:n
326:2
319:2
314:n
312:Q
307:3
304:Q
292:n
287:n
285:Q
261:n
259:Q
213:}
210:n
207:,
201:,
198:0
195:=
192:k
189:;
182:)
177:k
174:n
169:(
163:)
159:k
156:2
150:n
147:(
144:{
126:2
112:n
97:n
92:4
80:n
67:n
65:2
53:2
39:4
36:Q
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
