655:
1199:
849:
773:
580:
531:
483:
434:
153:
1229:
1121:
1023:
923:
692:
1091:
989:
889:
217:
1154:
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956:
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728:
385:
340:
309:
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108:
78:
1260:
Goldberg, N., Mattman, T. W., & Naimi, R. (2014). Many, many more intrinsically knotted graphs. Algebraic & Geometric
Topology, 14(3), 1801-1823.
353:, all of which are minor minimal with this property. The other six graphs have knotless embeddings. This shows that knotless graphs are not closed under
697:
862:
built on them can be embedded into 4-space. Hein van der Holst (2006) showed that the graphs in the
Heawood family are not 4-flat and have
863:
191:
899:. Van der Holst suggested that they might form the complete list of excluded minors for both the 4-flat graphs and the graphs with
776:
534:
354:
312:
43:
586:
486:
350:
183:
928:
This conjecture can be further motivated from structural similarities to other topologically defined graphs classes:
179:
1304:
596:
1273:
1202:
855:
187:
1159:
809:
733:
540:
491:
443:
394:
113:
1208:
1100:
1002:
902:
671:
1057:
961:
992:
868:
537:. The family consists of 58 graphs, all of which have 22 edges. The unique smallest member,
315:. The family consists of 20 graphs, all of which have 21 edges. The unique smallest member,
196:
1132:
1030:
934:
782:
706:
363:
318:
287:
257:
226:
86:
56:
1124:
168:
668:
Is the
Heawood family the complete list of excluded minors of the 4-flat graphs and for
1094:
896:
282:
172:
51:
1298:
343:
164:
28:
996:
892:
35:
42:
refers to either one of the following two related graph families generated via
859:
17:
159:
In either setting the members of the graph family are collectively known as
660:
1286:
592:
582:, has eight vertices. The unique largest member has 14 vertices.
390:
222:
182:. They contain the smallest known examples for graphs that are
1276:. Journal of Combinatorial Theory, Series B, 96(3), 388-404.
858:, i.e., graphs with the property that every 2-dimensional
1211:
1162:
1135:
1103:
1060:
1033:
1005:
964:
937:
905:
871:
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785:
736:
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599:
543:
494:
446:
397:
366:
342:, has seven vertices. The unique largest member, the
321:
290:
260:
229:
199:
116:
89:
59:
1201:
are conjectured to generate all excluded minors for
854:This graph family has significance in the study of
1223:
1193:
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1115:
1085:
1046:
1017:
983:
950:
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649:
574:
525:
477:
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379:
334:
303:
273:
242:
211:
147:
102:
72:
178:The Heawood families play a significant role in
31:, which is a member of the Heawood graph family.
8:
1287:Topological Symmetries of the Heawood family
1256:
1254:
1252:
1250:
1248:
644:
600:
50:the family of 20 graphs generated from the
1274:Graphs and obstructions in four dimensions
589:and are minor minimal with this property.
171:, which too is named after its member the
1210:
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1140:
1134:
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121:
115:
94:
88:
64:
58:
1268:
1266:
1244:
703:The Heawood family generated from both
698:(more unsolved problems in mathematics)
167:is a member. This is in analogy to the
83:the family of 78 graphs generated from
1285:Mellor, B., & Wilson, R. (2023).
650:{\displaystyle \{K_{7},K_{3,3,1,1}\}}
7:
851:-family. It consists of 78 graphs.
1289:. arXiv preprint arXiv:2311.08573.
891:. In particular, they are neither
387:-family are intrinsically chiral.
25:
1093:generate all excluded minors for
864:Colin de Verdière graph invariant
349:Only 14 out of the 20 graphs are
192:Colin de Verdière graph invariant
775:through repeated application of
533:through repeated application of
311:through repeated application of
663:Unsolved problem in mathematics
995:) are the excluded minors for
585:All graphs in this family are
485:-family is generated from the
281:-family is generated from the
1:
779:is the disjoint union of the
27:Not to be confused with the
1194:{\displaystyle K_{3,3,1,1}}
844:{\displaystyle K_{3,3,1,1}}
768:{\displaystyle K_{3,3,1,1}}
575:{\displaystyle K_{3,3,1,1}}
526:{\displaystyle K_{3,3,1,1}}
487:complete multipartite graph
478:{\displaystyle K_{3,3,1,1}}
429:{\displaystyle K_{3,3,1,1}}
148:{\displaystyle K_{3,3,1,1}}
1321:
1272:van der Holst, H. (2006).
1224:{\displaystyle \mu \leq 5}
1116:{\displaystyle \mu \leq 4}
1018:{\displaystyle \mu \leq 3}
918:{\displaystyle \mu \leq 5}
777:ΔY- and YΔ-transformations
687:{\displaystyle \mu \leq 5}
535:ΔY- and YΔ-transformations
355:ΔY- and YΔ-transformations
313:ΔY- and YΔ-transformations
44:ΔY- and YΔ-transformations
26:
1086:{\displaystyle K_{3,3,1}}
180:topological graph theory
984:{\displaystyle K_{3,3}}
1225:
1195:
1150:
1117:
1087:
1048:
1019:
985:
952:
919:
885:
884:{\displaystyle \mu =6}
845:
800:
769:
724:
688:
651:
576:
527:
479:
430:
381:
336:
305:
275:
244:
213:
212:{\displaystyle \mu =6}
149:
104:
74:
1226:
1196:
1151:
1149:{\displaystyle K_{7}}
1118:
1088:
1049:
1047:{\displaystyle K_{6}}
1020:
986:
953:
951:{\displaystyle K_{5}}
920:
886:
846:
801:
799:{\displaystyle K_{7}}
770:
725:
723:{\displaystyle K_{7}}
689:
652:
587:intrinsically knotted
577:
528:
480:
431:
382:
380:{\displaystyle K_{7}}
351:intrinsically knotted
337:
335:{\displaystyle K_{7}}
306:
304:{\displaystyle K_{7}}
276:
274:{\displaystyle K_{7}}
245:
243:{\displaystyle K_{7}}
214:
184:intrinsically knotted
150:
105:
103:{\displaystyle K_{7}}
75:
73:{\displaystyle K_{7}}
1209:
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1133:
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1031:
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597:
541:
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288:
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227:
197:
114:
87:
57:
360:All members of the
346:, has 14 vertices.
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572:
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332:
301:
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100:
70:
993:Kuratowski graphs
16:(Redirected from
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806:-family and the
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1125:Petersen family
1099:
1098:
1095:linkless graphs
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965:
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190:, or that have
186:, that are not
169:Petersen family
117:
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111:
90:
85:
84:
60:
55:
54:
32:
23:
22:
15:
12:
11:
5:
1318:
1316:
1308:
1307:
1305:Graph families
1297:
1296:
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1262:
1243:
1241:
1238:
1237:
1236:
1233:Heawood family
1220:
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389:
374:
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329:
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298:
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283:complete graph
268:
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237:
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221:
208:
205:
202:
173:Petersen graph
161:Heawood graphs
157:
156:
142:
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136:
133:
130:
127:
124:
120:
97:
93:
81:
67:
63:
52:complete graph
40:Heawood family
24:
18:Heawood graphs
14:
13:
10:
9:
6:
4:
3:
2:
1317:
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1288:
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1249:
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1239:
1234:
1218:
1215:
1212:
1204:
1203:4-flat graphs
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1183:
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1177:
1174:
1171:
1168:
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1110:
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1078:
1075:
1072:
1069:
1066:
1062:
1039:
1035:
1027:
1012:
1009:
1006:
998:
997:planar graphs
994:
976:
973:
970:
966:
943:
939:
931:
930:
929:
926:
912:
909:
906:
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894:
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872:
865:
861:
857:
856:4-flat graphs
852:
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830:
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757:
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409:
406:
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388:
372:
368:
358:
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347:
345:
344:Heawood graph
327:
323:
314:
296:
292:
284:
266:
262:
235:
231:
220:
206:
203:
200:
193:
189:
185:
181:
176:
174:
170:
166:
165:Heawood graph
162:
140:
137:
134:
131:
128:
125:
122:
118:
95:
91:
82:
65:
61:
53:
49:
48:
47:
45:
41:
37:
30:
29:Heawood graph
19:
1281:
1232:
927:
853:
702:
584:
439:
359:
348:
253:
177:
160:
158:
39:
36:graph theory
33:
1240:References
860:CW complex
1216:≤
1213:μ
1108:≤
1105:μ
1010:≤
1007:μ
910:≤
907:μ
873:μ
679:≤
676:μ
201:μ
163:, as the
38:the term
1299:Category
897:linkless
657:-family
436:-family
250:-family
893:planar
188:4-flat
1231:(the
1123:(the
991:(the
1205:and
1156:and
1097:and
1054:and
999:and
958:and
895:nor
730:and
593:The
440:The
391:The
254:The
223:The
110:and
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1301::
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219:.
175:.
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20:)
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