606:
590:
44:
367:
of the group of permutations swap one side of the bipartition for the other. As
Coxeter showed, any path of up to five edges in the Tutte–Coxeter graph is equivalent to any other such path by one such automorphism.
363:
graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set. In addition, the
426:
281:; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a).
333:
to its 15 edges, as described by
Coxeter (1958b), based on work by Sylvester (1844). Each vertex corresponds to an edge or a perfect matching, and connected vertices represent the
512:
549:
575:
453:
605:
483:
589:
203:
270:
943:
741:
295:
933:
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352:, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The
192:
383:
948:
463:
95:
85:
456:
377:
288:
260:
188:
633:
Brouwer, A. E.; Cohen, A. M.; and
Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
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105:
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115:
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129:
58:
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341:
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196:
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149:
139:
830:
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468:
326:
299:
927:
846:
778:
232:
28:
643:
303:
216:
159:
919:
680:
428:(there is an exceptional isomorphism between this group and the symmetric group
285:
248:
236:
212:
180:
172:
54:
17:
904:
43:
838:
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318:
256:
340:
Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a
909:
875:
858:
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710:
518:
vertices are either nonzero vectors, or isotropic 2-dimensional subspaces,
462:
Concretely, the Tutte-Coxeter graph can be defined from a 4-dimensional
664:
647:
770:
739:(1958b). "Twelve points in PG(5,3) with 95040 self-transformations".
791:"Elementary researches in the analysis of combinatorial aggregation"
790:
356:
of this group correspond to permuting the six vertices of the
291:
are known. The Tutte–Coxeter is one of the 13 such graphs.
455:). More specifically, it is the incidence graph of a
235:
with 30 vertices and 45 edges. As the unique smallest
557:
531:
491:
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434:
386:
168:
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104:
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84:
74:
64:
50:
36:
918:Exoo, G. "Rectilinear Drawings of Famous Graphs."
569:
543:
506:
477:
447:
420:
859:"The chords of the non-ruled quadric in PG(3,3)"
711:"The chords of the non-ruled quadric in PG(3,3)"
8:
697:Master Thesis, University of Tübingen, 2018
521:there is an edge between a nonzero vector
874:
726:
663:
556:
530:
498:
494:
493:
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470:
439:
433:
409:
405:
404:
394:
385:
525:and an isotropic 2-dimensional subspace
421:{\displaystyle Sp_{4}(\mathbb {F} _{2})}
681:"Rectilinear Drawings of Famous Graphs"
626:
585:
817:(1947). "A family of cubical graphs".
33:
372:The Tutte–Coxeter graph as a building
7:
695:Engineering Linear Layouts with SAT.
380:associated to the symplectic group
742:Proceedings of the Royal Society A
25:
615:of the Tutte–Coxeter graph is 3.
604:
599:of the Tutte–Coxeter graph is 2.
588:
507:{\displaystyle \mathbb {F} _{2}}
255:, and can be constructed as the
42:
314:The Tutte–Coxeter graph is the
310:Constructions and automorphisms
415:
400:
271:Cremona–Richmond configuration
204:Table of graphs and parameters
1:
337:between edges and matchings.
893:"3D Model of Tutte's 8-cage"
273:). The graph is named after
819:Proc. Cambridge Philos. Soc
965:
544:{\displaystyle W\subset V}
26:
944:Configurations (geometry)
839:10.1017/S0305004100023720
807:10.1080/14786444408644856
202:
41:
648:"Crossing Number Graphs"
27:Not to be confused with
464:symplectic vector space
289:distance-regular graphs
876:10.4153/CJM-1958-046-3
763:10.1098/rspa.1958.0184
728:10.4153/CJM-1958-047-0
571:
570:{\displaystyle v\in W}
545:
508:
479:
457:generalized quadrangle
449:
422:
261:generalized quadrangle
229:Cremona–Richmond graph
572:
546:
509:
480:
450:
448:{\displaystyle S_{6}}
423:
555:
529:
489:
469:
432:
384:
275:William Thomas Tutte
831:1947PCPS...43..459T
755:1958RSPSA.247..279C
652:Mathematica Journal
646:; Exoo, G. (2009).
365:outer automorphisms
354:inner automorphisms
335:incidence structure
221:Tutte–Coxeter graph
193:Distance-transitive
37:Tutte–Coxeter graph
934:1958 introductions
902:Weisstein, Eric W.
891:François Labelle.
665:10.3888/tmj.11.2-2
567:
541:
504:
475:
445:
418:
378:spherical building
376:This graph is the
321:connecting the 15
939:Individual graphs
749:(1250): 279–293.
737:Coxeter, H. S. M.
707:Coxeter, H. S. M.
478:{\displaystyle V}
323:perfect matchings
209:
208:
16:(Redirected from
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787:Sylvester, J. J.
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597:chromatic number
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279:H. S. M. Coxeter
225:Tutte eight-cage
189:Distance-regular
130:Chromatic number
59:H. S. M. Coxeter
46:
34:
21:
18:Tutte eight cage
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693:Wolz, Jessica;
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613:chromatic index
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551:if and only if
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342:symmetric graph
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296:crossing number
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140:Chromatic index
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32:
23:
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15:
12:
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5:
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949:Regular graphs
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885:External links
883:
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825:(4): 459–474.
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327:complete graph
325:of a 6-vertex
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300:book thickness
269:(known as the
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150:Book thickness
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350:automorphisms
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233:regular graph
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116:Automorphisms
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29:Coxeter graph
19:
908:
905:"Levi Graph"
866:
863:Can. J. Math
862:
855:Tutte, W. T.
822:
818:
815:Tutte, W. T.
798:
797:. Series 3.
794:
746:
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718:
715:Can. J. Math
714:
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514:as follows:
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304:queue number
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220:
217:graph theory
213:mathematical
210:
160:Queue number
869:: 481–483.
801:: 285–295.
721:: 484–488.
644:Pegg, E. T.
344:; it has a
249:Moore graph
243:8, it is a
237:cubic graph
181:Moore graph
120:1440 (Aut(S
55:W. T. Tutte
51:Named after
928:Categories
621:References
319:Levi graph
257:Levi graph
169:Properties
910:MathWorld
847:123505185
795:Phil. Mag
779:121676627
709:(1958a).
679:Exoo, G.
562:∈
536:⊂
316:bipartite
253:bipartite
215:field of
197:Bipartite
185:Symmetric
857:(1958).
789:(1844).
348:of 1440
284:All the
251:. It is
96:Diameter
66:Vertices
827:Bibcode
751:Bibcode
582:Gallery
294:It has
259:of the
231:is a 3-
211:In the
845:
777:
771:100667
769:
302:3 and
247:and a
219:, the
86:Radius
843:S2CID
775:S2CID
767:JSTOR
658:(2).
485:over
346:group
286:cubic
241:girth
173:Cubic
106:Girth
76:Edges
611:The
595:The
298:13,
277:and
245:cage
177:Cage
871:doi
835:doi
803:doi
759:doi
747:247
723:doi
660:doi
306:2.
239:of
227:or
223:or
930::
907:.
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865:.
861:.
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833:.
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793:.
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459:.
124:))
80:45
70:30
913:.
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873::
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837::
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805::
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761::
753::
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725::
683:.
668:.
662::
577:.
565:W
559:v
539:V
533:W
523:v
500:2
495:F
473:V
441:6
437:S
416:)
411:2
406:F
401:(
396:4
392:p
388:S
361:6
358:K
331:6
329:K
267:2
264:W
164:2
154:3
144:3
134:2
122:6
110:8
100:4
90:4
31:.
20:)
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