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1249:
Goldberg, N., Mattman, T. W., & Naimi, R. (2014). Many, many more intrinsically knotted graphs. Algebraic & Geometric
Topology, 14(3), 1801-1823.
342:, 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
686:
851:
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
852:
180:
888:. Van der Holst suggested that they might form the complete list of excluded minors for both the 4-flat graphs and the graphs with
765:
523:
343:
301:
32:
575:
475:
339:
172:
917:
This conjecture can be further motivated from structural similarities to other topologically defined graphs classes:
168:
1293:
585:
1262:
1191:
844:
176:
1148:
798:
722:
529:
480:
432:
383:
102:
1197:
1089:
991:
891:
660:
1046:
950:
981:
857:
526:. The family consists of 58 graphs, all of which have 22 edges. The unique smallest member,
304:. The family consists of 20 graphs, all of which have 21 edges. The unique smallest member,
185:
1121:
1019:
923:
771:
695:
352:
307:
276:
246:
215:
75:
45:
1113:
157:
657:
Is the
Heawood family the complete list of excluded minors of the 4-flat graphs and for
1083:
885:
271:
161:
40:
1287:
332:
153:
17:
985:
881:
24:
31:
refers to either one of the following two related graph families generated via
848:
148:
In either setting the members of the graph family are collectively known as
649:
1275:
581:
571:, has eight vertices. The unique largest member has 14 vertices.
379:
211:
171:. They contain the smallest known examples for graphs that are
1265:. Journal of Combinatorial Theory, Series B, 96(3), 388-404.
847:, i.e., graphs with the property that every 2-dimensional
1200:
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926:
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331:, has seven vertices. The unique largest member, the
310:
279:
249:
218:
188:
105:
78:
48:
1190:
are conjectured to generate all excluded minors for
843:This graph family has significance in the study of
1212:
1182:
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1104:
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1035:
1006:
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939:
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832:
787:
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563:
514:
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368:
323:
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262:
231:
200:
136:
91:
61:
167:The Heawood families play a significant role in
20:, which is a member of the Heawood graph family.
8:
1276:Topological Symmetries of the Heawood family
1245:
1243:
1241:
1239:
1237:
633:
589:
39:the family of 20 graphs generated from the
1263:Graphs and obstructions in four dimensions
578:and are minor minimal with this property.
160:, which too is named after its member the
1199:
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110:
104:
83:
77:
53:
47:
1257:
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1233:
692:The Heawood family generated from both
687:(more unsolved problems in mathematics)
156:is a member. This is in analogy to the
72:the family of 78 graphs generated from
1274:Mellor, B., & Wilson, R. (2023).
639:{\displaystyle \{K_{7},K_{3,3,1,1}\}}
7:
840:-family. It consists of 78 graphs.
1278:. arXiv preprint arXiv:2311.08573.
880:. In particular, they are neither
376:-family are intrinsically chiral.
14:
1082:generate all excluded minors for
853:Colin de Verdière graph invariant
338:Only 14 out of the 20 graphs are
181:Colin de Verdière graph invariant
764:through repeated application of
522:through repeated application of
300:through repeated application of
652:Unsolved problem in mathematics
984:) are the excluded minors for
574:All graphs in this family are
474:-family is generated from the
270:-family is generated from the
1:
768:is the disjoint union of the
16:Not to be confused with the
1183:{\displaystyle K_{3,3,1,1}}
833:{\displaystyle K_{3,3,1,1}}
757:{\displaystyle K_{3,3,1,1}}
564:{\displaystyle K_{3,3,1,1}}
515:{\displaystyle K_{3,3,1,1}}
476:complete multipartite graph
467:{\displaystyle K_{3,3,1,1}}
418:{\displaystyle K_{3,3,1,1}}
137:{\displaystyle K_{3,3,1,1}}
1310:
1261:van der Holst, H. (2006).
1213:{\displaystyle \mu \leq 5}
1105:{\displaystyle \mu \leq 4}
1007:{\displaystyle \mu \leq 3}
907:{\displaystyle \mu \leq 5}
766:ΔY- and YΔ-transformations
676:{\displaystyle \mu \leq 5}
524:ΔY- and YΔ-transformations
344:ΔY- and YΔ-transformations
302:ΔY- and YΔ-transformations
33:ΔY- and YΔ-transformations
15:
1075:{\displaystyle K_{3,3,1}}
169:topological graph theory
973:{\displaystyle K_{3,3}}
1214:
1184:
1139:
1106:
1076:
1037:
1008:
974:
941:
908:
874:
873:{\displaystyle \mu =6}
834:
789:
758:
713:
677:
640:
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516:
468:
419:
370:
325:
294:
264:
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202:
201:{\displaystyle \mu =6}
138:
93:
63:
1215:
1185:
1140:
1138:{\displaystyle K_{7}}
1107:
1077:
1038:
1036:{\displaystyle K_{6}}
1009:
975:
942:
940:{\displaystyle K_{5}}
909:
875:
835:
790:
788:{\displaystyle K_{7}}
759:
714:
712:{\displaystyle K_{7}}
678:
641:
576:intrinsically knotted
566:
517:
469:
420:
371:
369:{\displaystyle K_{7}}
340:intrinsically knotted
326:
324:{\displaystyle K_{7}}
295:
293:{\displaystyle K_{7}}
265:
263:{\displaystyle K_{7}}
234:
232:{\displaystyle K_{7}}
203:
173:intrinsically knotted
139:
94:
92:{\displaystyle K_{7}}
64:
62:{\displaystyle K_{7}}
1198:
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103:
76:
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349:All members of the
335:, has 14 vertices.
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561:
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134:
89:
59:
982:Kuratowski graphs
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795:-family and the
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1119:
1114:Petersen family
1088:
1087:
1084:linkless graphs
1050:
1045:
1044:
1023:
1018:
1017:
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954:
949:
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922:
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250:
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184:
183:
179:, or that have
175:, that are not
158:Petersen family
106:
101:
100:
79:
74:
73:
49:
44:
43:
21:
12:
11:
5:
1307:
1305:
1297:
1296:
1294:Graph families
1286:
1285:
1281:
1280:
1267:
1251:
1232:
1230:
1227:
1226:
1225:
1222:Heawood family
1209:
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1155:
1132:
1128:
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412:
409:
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287:
283:
272:complete graph
257:
253:
240:
226:
222:
210:
197:
194:
191:
162:Petersen graph
150:Heawood graphs
146:
145:
131:
128:
125:
122:
119:
116:
113:
109:
86:
82:
70:
56:
52:
41:complete graph
29:Heawood family
13:
10:
9:
6:
4:
3:
2:
1306:
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1292:
1291:
1289:
1277:
1271:
1268:
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1246:
1244:
1242:
1240:
1238:
1234:
1228:
1223:
1207:
1204:
1201:
1193:
1192:4-flat graphs
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1153:
1130:
1126:
1118:
1115:
1099:
1096:
1093:
1085:
1067:
1064:
1061:
1058:
1055:
1051:
1028:
1024:
1016:
1001:
998:
995:
987:
986:planar graphs
983:
965:
962:
959:
955:
932:
928:
920:
919:
918:
915:
901:
898:
895:
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883:
867:
864:
861:
854:
850:
846:
845:4-flat graphs
841:
825:
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819:
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767:
749:
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688:
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664:
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577:
572:
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541:
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492:
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485:
477:
459:
456:
453:
450:
447:
444:
441:
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410:
407:
404:
401:
398:
395:
392:
388:
377:
361:
357:
347:
345:
341:
336:
334:
333:Heawood graph
316:
312:
303:
285:
281:
273:
255:
251:
224:
220:
209:
195:
192:
189:
182:
178:
174:
170:
165:
163:
159:
155:
154:Heawood graph
151:
129:
126:
123:
120:
117:
114:
111:
107:
84:
80:
71:
54:
50:
42:
38:
37:
36:
34:
30:
26:
19:
18:Heawood graph
1270:
1221:
916:
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691:
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242:
166:
149:
147:
28:
25:graph theory
22:
1229:References
849:CW complex
1205:≤
1202:μ
1097:≤
1094:μ
999:≤
996:μ
899:≤
896:μ
862:μ
668:≤
665:μ
190:μ
152:, as the
27:the term
1288:Category
886:linkless
646:-family
425:-family
239:-family
882:planar
177:4-flat
1220:(the
1112:(the
980:(the
1194:and
1145:and
1086:and
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947:and
884:nor
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