29:
548:
759:
197:
282:
244:
523:
823:: he had conjectured that the complete graph has the smallest Ramsey number among all graphs with the same chromatic number, but Faudree and McKay (1993) showed
541:
635:
1061:
949:
319:
922:
128:
248:
1017:
583:
3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even
202:
978:
914:
67:
1021:
486:
1066:
420:
Given a vertex set of {1, 2, 3, …, v}, the edge set of the wheel graph can be represented in
121:
39:
906:
619:
421:
91:
8:
907:
612:
352:
51:
995:
366:
408:), which is formed by connecting a single vertex to all vertices of a cycle of length
945:
939:
918:
881:
480:
445:
291:
999:
987:
597:
580:
340:
101:
781:
944:(Corrected, enlarged republication. ed.). New York: Dover Pub. p. 56.
884:
436:, and have a unique planar embedding. More specifically, every wheel graph is a
830:
has Ramsey number 17 while the complete graph with the same chromatic number,
1055:
1013:
820:
802:
576:
816:
973:
433:
332:
328:
299:
801:-wheel by considering the outer cycle of the wheel, as well as all of its
853:
437:
344:
991:
768:
theory, two particularly important special classes of matroids are the
441:
424:
by {{1, 2}, {1, 3}, …, {1, v}, {2, 3}, {3, 4}, …, {v − 1, v}, {v, 2}}.
295:
889:
765:
28:
754:{\displaystyle P_{W_{n}}(x)=x((x-2)^{(n-1)}-(-1)^{n}(x-2)).}
837:, has Ramsey number 18. That is, for every 17-vertex graph
547:
536:
638:
489:
412:. The rest of this article uses the former notation.
251:
205:
131:
879:
753:
517:
276:
238:
191:
976:(1988), "On the euclidean dimension of a wheel",
1053:
1037:J. Combinatorial Math. and Combinatorial Comput.
1022:"A conjecture of Erdős and the Ramsey number
971:
815:supplied a counterexample to a conjecture of
1012:
271:
255:
233:
132:
852:as a subgraph, while neither the 17-vertex
415:
192:{\displaystyle \{2\cos(2k\pi /(n-1))^{1};}
909:Discrete Mathematics and Its Applications
448:. Every maximal planar graph, other than
339:is a graph formed by connecting a single
277:{\displaystyle \cup \{1\pm {\sqrt {n}}\}}
546:
937:
1054:
856:nor its complement contains a copy of
776:, both derived from wheel graphs. The
351:vertices can also be defined as the 1-
913:(7th ed.). McGraw-Hill. p.
904:
880:
797:-whirl matroid is derived from the
13:
611:is the only wheel graph that is a
14:
1078:
483:in the wheel graph and there are
239:{\displaystyle k=1,\ldots ,n-2\}}
16:Cycle graph plus universal vertex
551:The 7 cycles of the wheel graph
462:, contains as a subgraph either
33:Several examples of wheel graphs
27:
1006:
965:
931:
898:
873:
745:
742:
730:
721:
711:
703:
691:
687:
674:
671:
662:
656:
320:Table of graphs and parameters
177:
173:
161:
144:
1:
1062:Parametric families of graphs
866:
427:
394:to denote a wheel graph with
387:); other authors instead use
376:to denote a wheel graph with
941:Introduction to Graph Theory
938:Trudeau, Richard J. (1993).
7:
845:or its complement contains
10:
1083:
905:Rosen, Kenneth H. (2011).
518:{\displaystyle n^{2}-3n+3}
440:. They are self-dual: the
444:of any wheel graph is an
318:
305:
287:
120:
100:
90:
66:
50:
38:
26:
21:
979:Graphs and Combinatorics
615:in the Euclidean plane.
416:Set-builder construction
780:-wheel matroid is the
755:
604:≥ 6) is not perfect.
559:
519:
278:
240:
193:
805:, to be independent.
756:
550:
520:
369:. Some authors write
347:. A wheel graph with
343:to all vertices of a
279:
241:
194:
636:
620:chromatic polynomial
487:
422:set-builder notation
249:
203:
129:
622:of the wheel graph
613:unit distance graph
992:10.1007/BF01864150
882:Weisstein, Eric W.
751:
562:For odd values of
560:
515:
479:There is always a
274:
236:
189:
1018:McKay, Brendan D.
1014:Faudree, Ralph J.
951:978-0-486-67870-2
481:Hamiltonian cycle
432:Wheel graphs are
325:
324:
269:
1074:
1046:
1044:
1010:
1004:
1002:
969:
963:
962:
960:
958:
935:
929:
928:
912:
902:
896:
895:
894:
877:
760:
758:
757:
752:
729:
728:
707:
706:
655:
654:
653:
652:
598:chromatic number
581:chromatic number
539:
524:
522:
521:
516:
499:
498:
446:isomorphic graph
411:
407:
400:
393:
386:
379:
375:
365:
363:
350:
341:universal vertex
314:
283:
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245:
243:
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198:
196:
195:
190:
185:
184:
160:
115:
109:
102:Chromatic number
86:
78:
62:
46:
31:
19:
18:
1082:
1081:
1077:
1076:
1075:
1073:
1072:
1071:
1052:
1051:
1050:
1049:
1032:
1011:
1007:
972:Buckley, Fred;
970:
966:
956:
954:
952:
936:
932:
925:
903:
899:
878:
874:
869:
862:
851:
836:
829:
814:
792:
782:graphic matroid
724:
720:
690:
686:
648:
644:
643:
639:
637:
634:
633:
627:
610:
595:
574:
557:
535:
533:
494:
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468:
461:
454:
430:
418:
409:
402:
395:
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348:
313:
309:
298:
294:
264:
250:
247:
246:
204:
201:
200:
199:
180:
176:
156:
130:
127:
126:
113:
111:
107:
81:
79:
73:
56:
44:
34:
17:
12:
11:
5:
1080:
1070:
1069:
1064:
1048:
1047:
1030:
1005:
964:
950:
930:
924:978-0073383095
923:
897:
871:
870:
868:
865:
860:
849:
834:
827:
812:
803:spanning trees
788:
774:whirl matroids
770:wheel matroids
762:
761:
750:
747:
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741:
738:
735:
732:
727:
723:
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608:
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570:
555:
529:
514:
511:
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459:
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429:
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390:
372:
331:discipline of
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36:
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24:
23:
15:
9:
6:
4:
3:
2:
1079:
1068:
1067:Planar graphs
1065:
1063:
1060:
1059:
1057:
1042:
1038:
1034:
1029:
1025:
1019:
1015:
1009:
1001:
997:
993:
989:
985:
981:
980:
975:
974:Harary, Frank
968:
953:
947:
943:
942:
934:
926:
920:
916:
911:
910:
901:
892:
891:
886:
885:"Wheel Graph"
883:
876:
872:
864:
859:
855:
848:
844:
840:
833:
826:
822:
821:Ramsey theory
818:
811:
806:
804:
800:
796:
791:
787:
783:
779:
775:
771:
767:
748:
739:
736:
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717:
714:
708:
700:
697:
694:
683:
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677:
668:
665:
659:
649:
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640:
632:
631:
630:
628:
621:
616:
614:
607:
603:
600:4, and (when
599:
594:
590:
586:
582:
578:
577:perfect graph
573:
569:
565:
554:
549:
545:
543:
538:
532:
528:
512:
509:
506:
503:
500:
495:
491:
482:
477:
472:
465:
458:
451:
447:
443:
439:
435:
434:planar graphs
425:
423:
413:
405:
398:
384:
368:
361:
354:
346:
342:
338:
334:
330:
321:
317:
308:
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266:
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84:
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71:
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60:
55:
53:
49:
43:
41:
37:
30:
25:
20:
1040:
1036:
1027:
1023:
1008:
986:(1): 23–30,
983:
977:
967:
955:. Retrieved
940:
933:
908:
900:
888:
875:
857:
846:
842:
838:
831:
824:
809:
807:
798:
794:
793:, while the
789:
785:
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773:
769:
763:
623:
617:
605:
601:
592:
588:
584:
571:
567:
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561:
552:
530:
526:
478:
470:
463:
456:
449:
431:
419:
403:
396:
382:
359:
336:
333:graph theory
329:mathematical
326:
82:
74:
58:
854:Paley graph
784:of a wheel
442:planar dual
438:Halin graph
337:wheel graph
292:Hamiltonian
22:Wheel graph
1056:Categories
867:References
817:Paul Erdős
808:The wheel
629:is :
534:(sequence
525:cycles in
428:Properties
401:vertices (
380:vertices (
288:Properties
890:MathWorld
841:, either
737:−
715:−
709:−
698:−
681:−
501:−
296:Self-dual
262:±
253:∪
228:−
219:…
168:−
154:π
142:
1020:(1993),
1000:44596093
957:8 August
772:and the
353:skeleton
306:Notation
122:Spectrum
68:Diameter
40:Vertices
1043:: 23–31
766:matroid
540:in the
537:A002061
367:pyramid
364:)-gonal
327:In the
110:is even
998:
948:
921:
355:of an
300:Planar
116:is odd
77:> 4
996:S2CID
579:with
575:is a
345:cycle
112:3 if
106:4 if
92:Girth
80:1 if
72:2 if
52:Edges
45:n ≥ 4
959:2012
946:ISBN
919:ISBN
618:The
596:has
542:OEIS
335:, a
61:− 1)
988:doi
915:655
819:on
790:k+1
764:In
544:).
469:or
406:≥ 3
399:+ 1
385:≥ 4
362:– 1
139:cos
85:= 4
1058::
1041:13
1039:,
1035:,
1033:)"
1016:;
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917:.
887:.
863:.
587:,
566:,
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1026:(
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