95:
1325:
and their subgraphs. Each graph of this type has a vertex with at most two incident edges; one can 3-color any such graph by removing one such vertex, coloring the remaining graph recursively, and then adding back and coloring the removed vertex. Because the removed vertex has at most two edges, one
2454:
graph has
Hadwiger number greater than or equal to its chromatic number, so the Hadwiger conjecture holds for random graphs with high probability; more precisely, the Hadwiger number is with high probability proportional
2394:
2936:
1868:
1796:
667:(which is required for minors). Hadwiger's conjecture states that there exists a different way of properly edge contracting sets of vertices to single vertices, producing a complete
831:
761:
730:
2276:
1982:
1940:
1082:
2493:
2026:
241:
206:
2429:
1111:
3195:
3127:
2635:
2532:
2183:
2149:
1705:
1672:
1552:
1582:
3805:
3162:
3094:
2989:
2870:
2839:
2768:
2605:
2119:
2053:
1898:
1639:
1480:
1445:
1384:
1319:
1259:
1198:
967:
864:
794:
694:
636:
506:
380:
163:
2697:
2666:
900:
2237:
2211:
1612:
1514:
1418:
1353:
1290:
1224:
1171:
554:
3227:
2960:
2812:
2792:
2741:
2721:
2575:
2554:
2450:
2322:
2299:
2090:
1802:
algorithm that removes this low-degree vertex, colors the remaining graph, and then adds back the removed vertex and colors it, will color the given graph with
1728:
1136:
1036:
1009:
987:
940:
920:
660:
605:
581:
528:
477:
425:
400:
346:
317:
297:
277:
132:
74:
53:
98:
A graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case
1526:, a three-dimensional analogue of planar graphs, have chromatic number at most five. Due to this result, the conjecture is known to be true
352:
to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single vertex produces a
20:
4127:
2331:
1420:
is actually equivalent to the four color theorem and therefore we now know it to be true. As Wagner showed, every graph that has no
88:
2814:, each of which is two-colored, such that each pair of subtrees is connected by a monochromatic edge. Although graphs with no odd
2876:
3511:
1805:
1733:
2579:
clique minor. However, the maximum list chromatic number of planar graphs is 5, not 4, so the extension fails already for
663:
because there is (by definition) no edge between any two vertices in the same color class, thus the contraction is not an
3971:
3416:
834:
3976:
4011:
3870:
3780:
Norin, Sergey; Postle, Luke; Song, Zi-Xia (2023), "Breaking the degeneracy barrier for coloring graphs with no
3440:
799:
325:
2845:, a similar upper bound holds for them as it does for the standard Hadwiger conjecture: a graph with no odd
1873:
In the 1980s, Alexander V. Kostochka and Andrew
Thomason both independently proved that every graph with no
4041:
3878:
3874:
3749:
3573:
2069:
1391:
1173:
is trivial: a graph requires more than one color if and only if it has an edge, and that edge is itself a
4117:
805:
735:
704:
4072:
Yu, Xingxing; Zickfeld, Florian (2006), "Reducing HajĂłs' 4-coloring conjecture to 4-connected graphs",
3569:
3355:
2255:
1043:
1945:
1903:
2431:
vertices. In this context, it is worth noting that the probability also approaches one that a random
1987:
1322:
2460:
1455:, and each of these pieces can be 4-colored independently of each other, so the 4-colorability of a
3506:, Graduate Texts in Mathematics, vol. 173 (5th ed.), Springer, Berlin, pp. 183–186,
214:
176:
3359:
4112:
3809:
3709:
Kostochka, A. V. (1984), "Lower bound of the
Hadwiger number of graphs by their average degree",
2401:
3745:
3660:
3616:
3130:
1087:
249:
and is considered to be one of the most important and challenging open problems in the field.
3167:
3099:
2614:
2511:
2162:
2128:
1677:
1644:
1531:
3462:
Catlin, P. A. (1979), "HajĂłs's graph-colouring conjecture: variations and counterexamples",
1561:
459:(each merging the two endpoints of some edge into a single supervertex) that brings a graph
4122:
4095:
4034:
3999:
3956:
3920:
3840:
3783:
3773:
3730:
3653:
3521:
3140:
3072:
2994:
2967:
2848:
2817:
2746:
2583:
2097:
2031:
1876:
1617:
1458:
1423:
1362:
1297:
1237:
1231:
1176:
945:
842:
767:
672:
614:
484:
403:
358:
141:
82:
2673:
2642:
876:
8:
2216:
2190:
1591:
1493:
1397:
1387:
1359:: for, if the conjecture is true, every graph requiring five or more colors would have a
1332:
1269:
1203:
1150:
533:
349:
135:
1326:
of the three colors will always be available to color it when the vertex is added back.
609:
contracting each color class of the coloring to a single vertex will produce a complete
4061:
3960:
3924:
3882:
3818:
3734:
3698:
3554:
3532:
3489:
3400:
3389:
3371:
3212:
2945:
2797:
2777:
2726:
2706:
2560:
2539:
2435:
2307:
2284:
2075:
1713:
1523:
1356:
1121:
1021:
994:
972:
925:
905:
645:
590:
566:
513:
462:
432:
410:
385:
331:
302:
282:
262:
246:
117:
59:
38:
3430:
4065:
4025:
3964:
3507:
3476:
3454:
2065:
560:
3738:
3702:
3393:
4081:
4053:
4020:
3985:
3944:
3928:
3908:
3828:
3761:
3718:
3690:
3639:
3589:
3558:
3546:
3471:
3449:
3425:
3381:
1264:
In the same paper in which he introduced the conjecture, Hadwiger proved its truth
1114:
664:
456:
257:
169:
33:
3851:
1452:
4091:
4030:
3995:
3952:
3916:
3836:
3769:
3726:
3649:
3517:
1799:
1519:
1227:
871:
322:
1614:, some partial results are known: every 7-chromatic graph must contain either a
1482:-minor-free graph follows from the 4-colorability of each of the planar pieces.
4086:
3644:
3593:
3230:
3013:
and announced to be proved in 2001 by
Robertson, Sanders, Seymour, and Thomas.
3006:
452:
353:
253:
78:
3948:
3832:
3765:
3694:
732:
denotes the family of graphs having the property that all minors of graphs in
4106:
3899:
3600:
3537:
3002:
2503:
436:
3935:
Thomason, Andrew (1984), "An extremal function for contractions of graphs",
3528:
3408:
3404:
1518:
also using the four color theorem; their paper with this proof won the 1994
4006:
3990:
3847:
2842:
2247:
319:
107:
3010:
2998:
1984:
colors. A sequence of improvements to this bound have led to a proof of
165:
837:. Hadwiger's conjecture is that this set consists of a single forbidden
4057:
3912:
3722:
3565:
3550:
1448:
24:
94:
2072:
rather than minors: that is, that every graph with chromatic number
3823:
3494:
1040:
The
Hadwiger conjecture can be stated in the simple algebraic form
3376:
3385:
3041:
3039:
3037:
2068:
conjectured that
Hadwiger's conjecture could be strengthened to
443:
call it "one of the deepest unsolved problems in graph theory."
3937:
Mathematical
Proceedings of the Cambridge Philosophical Society
2389:{\displaystyle \geq ({\tfrac {1}{2}}-\varepsilon )n/\log _{2}n}
2303:
goes to infinity, the probability approaches one that a random
3577:
3486:
Reducing linear
Hadwiger's conjecture to coloring small graphs
3034:
455:
of the form stated above) is that, if there is no sequence of
2962:, it may be possible to prove the existence of larger minors
1226:
is also easy: the graphs requiring three colors are the non-
640:
However, this contraction process does not produce a minor
3303:
2931:{\textstyle \chi (G)=O{\bigl (}k{\sqrt {\log k}}{\bigr )}}
698:
in such a way that all the contracted sets are connected.
3603:(1943), "Ăśber eine Klassifikation der Streckenkomplexe",
3051:
2502:
asked whether
Hadwiger's conjecture could be extended to
3564:
3321:
3502:
Diestel, Reinhard (2017), "7.3 Hadwiger's conjecture",
3438:
Borowiecki, Mieczyslaw (1993), "Research problem 172",
1863:{\textstyle O{\bigl (}h(G){\sqrt {\log h(G)}}{\bigr )}}
1791:{\textstyle O{\bigl (}h(G){\sqrt {\log h(G)}}{\bigr )}}
3869:
3752:(1985), "A note on spatial representation of graphs",
3409:"Hadwiger's conjecture is true for almost every graph"
2879:
2463:
2404:
2342:
2157:
found counterexamples to this strengthened conjecture
1948:
1906:
1808:
1736:
1485:
989:(or equivalently can be obtained by contracting edges
431:
This conjecture, a far-reaching generalization of the
4044:(1937), "Ăśber eine Eigenschaft der ebenen Komplexe",
3786:
3275:
3215:
3170:
3143:
3102:
3075:
2970:
2948:
2851:
2820:
2800:
2780:
2774:. Such a structure can be represented as a family of
2749:
2729:
2709:
2676:
2645:
2617:
2586:
2563:
2542:
2514:
2438:
2334:
2310:
2287:
2258:
2219:
2193:
2165:
2131:
2100:
2078:
2034:
1990:
1879:
1716:
1680:
1647:
1620:
1594:
1564:
1534:
1496:
1461:
1426:
1400:
1365:
1335:
1300:
1272:
1240:
1206:
1179:
1153:
1124:
1090:
1046:
1024:
997:
975:
948:
928:
908:
879:
845:
808:
770:
738:
707:
675:
648:
617:
593:
569:
536:
516:
487:
465:
413:
388:
361:
334:
305:
285:
265:
217:
179:
144:
120:
62:
41:
3236:
1142:
3754:
3578:"On the odd-minor variant of Hadwiger's conjecture"
3399:
3248:
3045:
1234:, which can be contracted to a 3-cycle, that is, a
451:An equivalent form of the Hadwiger conjecture (the
440:
3799:
3663:; Toft, Bjarne (2005), "Any 7-chromatic graph has
3221:
3189:
3156:
3121:
3088:
3022:
2983:
2954:
2930:
2864:
2833:
2806:
2786:
2762:
2735:
2715:
2691:
2660:
2629:
2599:
2569:
2548:
2526:
2487:
2444:
2423:
2388:
2316:
2293:
2270:
2231:
2205:
2177:
2143:
2113:
2084:
2047:
2020:
1976:
1934:
1892:
1862:
1790:
1722:
1699:
1666:
1633:
1606:
1576:
1546:
1508:
1474:
1451:into pieces that are either planar or an 8-vertex
1439:
1412:
1378:
1347:
1313:
1284:
1253:
1218:
1192:
1165:
1130:
1105:
1076:
1030:
1003:
981:
961:
934:
914:
894:
858:
825:
788:
755:
724:
688:
654:
630:
599:
575:
548:
522:
500:
471:
419:
394:
374:
340:
311:
291:
271:
235:
200:
157:
126:
68:
47:
3659:
3134:
2703:Gerards and Seymour conjectured that every graph
16:Unproven generalization of the four-color theorem
4104:
3331:
2246:observed that HajĂłs' conjecture fails badly for
3852:"Book Review: The Colossal Book of Mathematics"
3779:
3527:
3270:
2243:
26:
3744:
3483:
3353:
3309:
3266:
3057:
3974:(1994), "Every planar graph is 5-choosable",
3619:(2009), "Note on coloring graphs without odd-
3615:
3325:
2923:
2900:
1855:
1814:
1798:incident edges, from which it follows that a
1783:
1742:
4009:(1993), "List colourings of planar graphs",
3859:Notices of the American Mathematical Society
2639:there exist graphs whose Hadwiger number is
2398:and that its largest clique subdivision has
21:Hadwiger conjecture (combinatorial geometry)
1230:, and every non-bipartite graph has an odd
348:such that each subgraph is connected by an
4071:
3437:
3360:"Disproof of the list Hadwiger conjecture"
3281:
2499:
245:The conjecture is a generalization of the
4085:
4074:Journal of Combinatorial Theory, Series B
4024:
3989:
3970:
3822:
3708:
3643:
3582:Journal of Combinatorial Theory, Series B
3493:
3484:Delcourt, Michelle; Postle, Luke (2021),
3475:
3464:Journal of Combinatorial Theory, Series B
3453:
3429:
3375:
3297:
3242:
3206:
3934:
3599:
3254:
2280:in the limit as the number of vertices,
833:can be characterized by a finite set of
93:
3501:
3028:
2536:every graph with list chromatic number
102: = 4 of Hadwiger's conjecture
89:(more unsolved problems in mathematics)
4105:
4040:
3605:Vierteljschr. Naturforsch. Ges. ZĂĽrich
3535:(1981), "On the conjecture of HajĂłs",
3461:
2154:
1486:Robertson, Seymour & Thomas (1993)
4005:
3293:
2092:contains a subdivision of a complete
3846:
3337:
3364:Electronic Journal of Combinatorics
3046:Bollobás, Catlin & Erdős (1980)
1522:. It follows from their proof that
446:
441:Bollobás, Catlin & Erdős (1980)
13:
2060:
826:{\displaystyle {\mathcal {F}}_{k}}
812:
756:{\displaystyle {\mathcal {F}}_{k}}
742:
725:{\displaystyle {\mathcal {F}}_{k}}
711:
299:or more colors, then one can find
14:
4139:
4128:Unsolved problems in graph theory
3417:European Journal of Combinatorics
3137:proved the existence of either a
2271:{\displaystyle \varepsilon >0}
2028:-colorability for graphs without
1977:{\textstyle O(k{\sqrt {\log k}})}
1935:{\textstyle O(k{\sqrt {\log k}})}
1143:Special cases and partial results
1077:{\displaystyle \chi (G)\leq h(G)}
530:must have a vertex coloring with
2942:By imposing extra conditions on
2668:and whose list chromatic number
1329:The truth of the conjecture for
3977:Journal of Combinatorial Theory
3632:Journal of Combinatorial Theory
3315:
3287:
3271:Norin, Postle & Song (2023)
3135:Kawarabayashi & Toft (2005)
2488:{\textstyle n/{\sqrt {\log n}}}
2021:{\displaystyle O(k\log \log k)}
439:in 1943 and is still unsolved.
27:Unsolved problem in mathematics
3354:Barát, János; Joret, Gwenaël;
3310:Barát, Joret & Wood (2011)
3260:
3200:
3063:
2889:
2883:
2418:
2408:
2359:
2338:
2015:
1994:
1971:
1952:
1942:and can thus be colored using
1929:
1910:
1848:
1842:
1828:
1822:
1776:
1770:
1756:
1750:
1100:
1094:
1071:
1065:
1056:
1050:
942:of the largest complete graph
889:
883:
783:
771:
189:
183:
1:
3431:10.1016/s0195-6698(80)80001-1
3346:
3001:requiring four colors in any
2244:Erdős & Fajtlowicz (1981)
1394:proved in 1937 that the case
236:{\displaystyle 1\leq t\leq 6}
201:{\displaystyle \chi (G)<t}
4026:10.1016/0012-365X(93)90579-I
3477:10.1016/0095-8956(79)90062-5
3455:10.1016/0012-365X(93)90557-A
3267:Delcourt & Postle (2021)
3058:Nešetřil & Thomas (1985)
2794:vertex-disjoint subtrees of
2609:graphs. More generally, for
1556:but it remains unsolved for
1524:linklessly embeddable graphs
1447:minor can be decomposed via
7:
3883:"Hadwiger's conjecture for
3229:in this expression invokes
3009:as a minor, conjectured by
2872:minor has chromatic number
2424:{\textstyle O({\sqrt {n}})}
210:It is known to be true for
10:
4144:
4087:10.1016/j.jctb.2005.10.001
3817:, Paper No. 109020, 23pp,
3645:10.1016/j.jctb.2008.12.001
3594:10.1016/j.jctb.2008.03.006
3069:The existence of either a
2841:minor are not necessarily
2123:HajĂłs' conjecture is true
1730:has a vertex with at most
18:
3949:10.1017/S0305004100061521
3833:10.1016/j.aim.2023.109020
3695:10.1007/s00493-005-0019-1
1900:minor has average degree
1015:contraction clique number
800:Robertson–Seymour theorem
798:then it follows from the
3576:; Vetta, Adrian (2006),
3282:Yu & Zickfeld (2006)
3016:
1106:{\displaystyle \chi (G)}
1013:It is also known as the
3810:Advances in Mathematics
3661:Kawarabayashi, Ken-ichi
3617:Kawarabayashi, Ken-ichi
3190:{\displaystyle K_{4,4}}
3122:{\displaystyle K_{3,5}}
2630:{\displaystyle t\geq 1}
2527:{\displaystyle k\leq 4}
2178:{\displaystyle k\geq 7}
2144:{\displaystyle k\leq 4}
1700:{\displaystyle K_{3,5}}
1667:{\displaystyle K_{4,4}}
1547:{\displaystyle k\leq 6}
252:In more detail, if all
3991:10.1006/jctb.1994.1062
3801:
3223:
3191:
3158:
3131:Ken-ichi Kawarabayashi
3123:
3090:
2985:
2956:
2932:
2866:
2835:
2808:
2788:
2764:
2737:
2723:with chromatic number
2717:
2693:
2662:
2631:
2601:
2571:
2550:
2528:
2489:
2446:
2425:
2390:
2318:
2295:
2272:
2233:
2207:
2179:
2145:
2115:
2086:
2049:
2022:
1978:
1936:
1894:
1864:
1792:
1724:
1701:
1668:
1635:
1608:
1578:
1577:{\displaystyle k>6}
1548:
1510:
1488:proved the conjecture
1476:
1441:
1414:
1380:
1349:
1323:series–parallel graphs
1315:
1286:
1255:
1220:
1194:
1167:
1132:
1107:
1078:
1032:
1005:
983:
963:
936:
916:
896:
860:
827:
790:
757:
726:
690:
656:
632:
601:
577:
550:
524:
502:
473:
421:
396:
376:
342:
313:
293:
273:
237:
202:
159:
128:
103:
70:
49:
32:Does every graph with
4046:Mathematische Annalen
3802:
3800:{\displaystyle K_{t}}
3224:
3192:
3159:
3157:{\displaystyle K_{7}}
3124:
3091:
3089:{\displaystyle K_{7}}
2986:
2984:{\displaystyle K_{k}}
2957:
2933:
2867:
2865:{\displaystyle K_{k}}
2836:
2834:{\displaystyle K_{k}}
2809:
2789:
2765:
2763:{\displaystyle K_{k}}
2743:has a complete graph
2738:
2718:
2694:
2663:
2632:
2602:
2600:{\displaystyle K_{5}}
2572:
2551:
2529:
2490:
2447:
2426:
2391:
2319:
2296:
2273:
2234:
2208:
2180:
2146:
2116:
2114:{\displaystyle K_{k}}
2087:
2050:
2048:{\displaystyle K_{k}}
2023:
1979:
1937:
1895:
1893:{\displaystyle K_{k}}
1865:
1793:
1725:
1702:
1669:
1636:
1634:{\displaystyle K_{7}}
1609:
1579:
1549:
1511:
1477:
1475:{\displaystyle K_{5}}
1442:
1440:{\displaystyle K_{5}}
1415:
1381:
1379:{\displaystyle K_{5}}
1350:
1316:
1314:{\displaystyle K_{4}}
1287:
1256:
1254:{\displaystyle K_{3}}
1221:
1195:
1193:{\displaystyle K_{2}}
1168:
1133:
1108:
1079:
1033:
1006:
984:
964:
962:{\displaystyle K_{k}}
937:
917:
897:
861:
859:{\displaystyle K_{k}}
828:
791:
789:{\displaystyle (k-1)}
758:
727:
691:
689:{\displaystyle K_{k}}
657:
633:
631:{\displaystyle K_{k}}
602:
578:
551:
525:
503:
501:{\displaystyle K_{k}}
474:
422:
397:
377:
375:{\displaystyle K_{k}}
343:
314:
294:
274:
238:
203:
160:
158:{\displaystyle K_{t}}
129:
97:
71:
50:
4012:Discrete Mathematics
3784:
3441:Discrete Mathematics
3326:Kawarabayashi (2009)
3322:Geelen et al. (2006)
3213:
3168:
3141:
3100:
3073:
2968:
2946:
2877:
2849:
2818:
2798:
2778:
2747:
2727:
2707:
2692:{\displaystyle 4t+1}
2674:
2661:{\displaystyle 3t+1}
2643:
2615:
2584:
2561:
2540:
2512:
2461:
2436:
2402:
2332:
2326:graph has chromatic
2308:
2285:
2256:
2217:
2191:
2163:
2129:
2098:
2076:
2032:
1988:
1946:
1904:
1877:
1806:
1734:
1714:
1678:
1645:
1618:
1592:
1562:
1532:
1494:
1459:
1424:
1398:
1386:minor and would (by
1363:
1333:
1298:
1270:
1238:
1204:
1177:
1151:
1122:
1088:
1044:
1022:
995:
973:
946:
926:
906:
895:{\displaystyle h(G)}
877:
843:
806:
768:
736:
705:
673:
646:
615:
591:
567:
534:
514:
485:
463:
411:
386:
359:
332:
303:
283:
263:
215:
177:
142:
118:
60:
39:
3533:Fajtlowicz, Siemion
3129:minor was shown by
2993:One example is the
2232:{\displaystyle k=6}
2206:{\displaystyle k=5}
1607:{\displaystyle k=7}
1509:{\displaystyle k=6}
1413:{\displaystyle k=5}
1348:{\displaystyle k=5}
1294:The graphs with no
1285:{\displaystyle k=4}
1219:{\displaystyle k=3}
1166:{\displaystyle k=2}
969:that is a minor of
549:{\displaystyle k-1}
112:Hadwiger conjecture
4118:Graph minor theory
4058:10.1007/BF01594196
3972:Thomassen, Carsten
3913:10.1007/BF01202354
3797:
3766:10338.dmlcz/106404
3746:Nešetřil, Jaroslav
3723:10.1007/BF02579141
3551:10.1007/BF02579269
3219:
3187:
3154:
3119:
3086:
2981:
2952:
2928:
2862:
2831:
2804:
2784:
2760:
2733:
2713:
2689:
2658:
2627:
2597:
2567:
2546:
2524:
2485:
2442:
2421:
2386:
2351:
2314:
2291:
2268:
2229:
2203:
2175:
2141:
2111:
2082:
2045:
2018:
1974:
1932:
1890:
1860:
1788:
1720:
1697:
1664:
1631:
1604:
1574:
1544:
1506:
1472:
1437:
1410:
1376:
1357:four color theorem
1345:
1311:
1282:
1251:
1216:
1190:
1163:
1128:
1103:
1074:
1028:
1001:
979:
959:
932:
912:
892:
856:
823:
786:
753:
722:
686:
652:
628:
597:
573:
546:
520:
498:
469:
433:four-color problem
417:
392:
372:
338:
309:
289:
269:
247:four-color theorem
233:
198:
155:
124:
104:
66:
45:
3568:; Gerards, Bert;
3513:978-3-662-57560-4
3403:; Catlin, P. A.;
3222:{\displaystyle O}
2955:{\displaystyle G}
2919:
2807:{\displaystyle G}
2787:{\displaystyle k}
2736:{\displaystyle k}
2716:{\displaystyle G}
2570:{\displaystyle k}
2549:{\displaystyle k}
2500:Borowiecki (1993)
2483:
2445:{\displaystyle n}
2416:
2350:
2317:{\displaystyle n}
2294:{\displaystyle n}
2085:{\displaystyle k}
1969:
1927:
1851:
1779:
1723:{\displaystyle G}
1131:{\displaystyle G}
1031:{\displaystyle G}
1004:{\displaystyle G}
982:{\displaystyle G}
935:{\displaystyle k}
915:{\displaystyle G}
655:{\displaystyle G}
600:{\displaystyle G}
576:{\displaystyle k}
523:{\displaystyle G}
472:{\displaystyle G}
457:edge contractions
420:{\displaystyle G}
395:{\displaystyle k}
341:{\displaystyle G}
312:{\displaystyle k}
292:{\displaystyle k}
272:{\displaystyle G}
127:{\displaystyle G}
69:{\displaystyle k}
48:{\displaystyle k}
4135:
4098:
4089:
4068:
4037:
4028:
4019:(1–3): 215–219,
4002:
3993:
3967:
3931:
3896:
3891:
3866:
3856:
3843:
3826:
3806:
3804:
3803:
3798:
3796:
3795:
3776:
3741:
3705:
3680:
3671:
3656:
3647:
3629:
3612:
3596:
3561:
3524:
3498:
3497:
3480:
3479:
3458:
3457:
3448:(1–3): 235–236,
3434:
3433:
3413:
3396:
3379:
3341:
3335:
3329:
3319:
3313:
3307:
3301:
3298:Thomassen (1994)
3291:
3285:
3279:
3273:
3264:
3258:
3252:
3246:
3243:Kostochka (1984)
3240:
3234:
3228:
3226:
3225:
3220:
3207:Kostochka (1984)
3204:
3198:
3196:
3194:
3193:
3188:
3186:
3185:
3163:
3161:
3160:
3155:
3153:
3152:
3128:
3126:
3125:
3120:
3118:
3117:
3095:
3093:
3092:
3087:
3085:
3084:
3067:
3061:
3055:
3049:
3043:
3032:
3026:
2992:
2990:
2988:
2987:
2982:
2980:
2979:
2961:
2959:
2958:
2953:
2939:
2937:
2935:
2934:
2929:
2927:
2926:
2920:
2909:
2904:
2903:
2871:
2869:
2868:
2863:
2861:
2860:
2840:
2838:
2837:
2832:
2830:
2829:
2813:
2811:
2810:
2805:
2793:
2791:
2790:
2785:
2769:
2767:
2766:
2761:
2759:
2758:
2742:
2740:
2739:
2734:
2722:
2720:
2719:
2714:
2700:
2698:
2696:
2695:
2690:
2667:
2665:
2664:
2659:
2638:
2636:
2634:
2633:
2628:
2608:
2606:
2604:
2603:
2598:
2596:
2595:
2578:
2576:
2574:
2573:
2568:
2555:
2553:
2552:
2547:
2535:
2533:
2531:
2530:
2525:
2496:
2494:
2492:
2491:
2486:
2484:
2473:
2471:
2453:
2451:
2449:
2448:
2443:
2430:
2428:
2427:
2422:
2417:
2412:
2397:
2395:
2393:
2392:
2387:
2379:
2378:
2369:
2352:
2343:
2325:
2323:
2321:
2320:
2315:
2302:
2300:
2298:
2297:
2292:
2279:
2277:
2275:
2274:
2269:
2242:
2238:
2236:
2235:
2230:
2212:
2210:
2209:
2204:
2186:
2184:
2182:
2181:
2176:
2152:
2150:
2148:
2147:
2142:
2122:
2120:
2118:
2117:
2112:
2110:
2109:
2091:
2089:
2088:
2083:
2057:
2054:
2052:
2051:
2046:
2044:
2043:
2027:
2025:
2024:
2019:
1983:
1981:
1980:
1975:
1970:
1959:
1941:
1939:
1938:
1933:
1928:
1917:
1899:
1897:
1896:
1891:
1889:
1888:
1869:
1867:
1866:
1861:
1859:
1858:
1852:
1832:
1818:
1817:
1797:
1795:
1794:
1789:
1787:
1786:
1780:
1760:
1746:
1745:
1729:
1727:
1726:
1721:
1706:
1704:
1703:
1698:
1696:
1695:
1673:
1671:
1670:
1665:
1663:
1662:
1641:minor or both a
1640:
1638:
1637:
1632:
1630:
1629:
1613:
1611:
1610:
1605:
1585:
1583:
1581:
1580:
1575:
1555:
1553:
1551:
1550:
1545:
1517:
1515:
1513:
1512:
1507:
1481:
1479:
1478:
1473:
1471:
1470:
1446:
1444:
1443:
1438:
1436:
1435:
1419:
1417:
1416:
1411:
1390:) be nonplanar.
1388:Wagner's theorem
1385:
1383:
1382:
1377:
1375:
1374:
1354:
1352:
1351:
1346:
1320:
1318:
1317:
1312:
1310:
1309:
1293:
1291:
1289:
1288:
1283:
1260:
1258:
1257:
1252:
1250:
1249:
1228:bipartite graphs
1225:
1223:
1222:
1217:
1200:minor. The case
1199:
1197:
1196:
1191:
1189:
1188:
1172:
1170:
1169:
1164:
1139:
1137:
1135:
1134:
1129:
1115:chromatic number
1112:
1110:
1109:
1104:
1083:
1081:
1080:
1075:
1039:
1037:
1035:
1034:
1029:
1012:
1010:
1008:
1007:
1002:
988:
986:
985:
980:
968:
966:
965:
960:
958:
957:
941:
939:
938:
933:
921:
919:
918:
913:
901:
899:
898:
893:
867:
865:
863:
862:
857:
855:
854:
835:forbidden minors
832:
830:
829:
824:
822:
821:
816:
815:
797:
795:
793:
792:
787:
762:
760:
759:
754:
752:
751:
746:
745:
731:
729:
728:
723:
721:
720:
715:
714:
697:
695:
693:
692:
687:
685:
684:
665:edge contraction
662:
661:
659:
658:
653:
639:
637:
635:
634:
629:
627:
626:
608:
606:
604:
603:
598:
584:
582:
580:
579:
574:
555:
553:
552:
547:
529:
527:
526:
521:
509:
507:
505:
504:
499:
497:
496:
479:to the complete
478:
476:
475:
470:
447:Equivalent forms
428:
426:
424:
423:
418:
401:
399:
398:
393:
381:
379:
378:
373:
371:
370:
347:
345:
344:
339:
318:
316:
315:
310:
298:
296:
295:
290:
278:
276:
275:
270:
258:undirected graph
254:proper colorings
244:
242:
240:
239:
234:
209:
207:
205:
204:
199:
170:chromatic number
164:
162:
161:
156:
154:
153:
133:
131:
130:
125:
77:
75:
73:
72:
67:
54:
52:
51:
46:
34:chromatic number
28:
4143:
4142:
4138:
4137:
4136:
4134:
4133:
4132:
4103:
4102:
4101:
3894:
3890:
3884:
3871:Robertson, Neil
3854:
3791:
3787:
3785:
3782:
3781:
3679:
3673:
3670:
3664:
3628:
3620:
3514:
3411:
3349:
3344:
3336:
3332:
3320:
3316:
3308:
3304:
3292:
3288:
3280:
3276:
3265:
3261:
3255:Thomason (1984)
3253:
3249:
3241:
3237:
3214:
3211:
3210:
3205:
3201:
3175:
3171:
3169:
3166:
3165:
3148:
3144:
3142:
3139:
3138:
3107:
3103:
3101:
3098:
3097:
3080:
3076:
3074:
3071:
3070:
3068:
3064:
3056:
3052:
3044:
3035:
3027:
3023:
3019:
2975:
2971:
2969:
2966:
2965:
2963:
2947:
2944:
2943:
2922:
2921:
2908:
2899:
2898:
2878:
2875:
2874:
2873:
2856:
2852:
2850:
2847:
2846:
2825:
2821:
2819:
2816:
2815:
2799:
2796:
2795:
2779:
2776:
2775:
2754:
2750:
2748:
2745:
2744:
2728:
2725:
2724:
2708:
2705:
2704:
2675:
2672:
2671:
2669:
2644:
2641:
2640:
2616:
2613:
2612:
2610:
2591:
2587:
2585:
2582:
2581:
2580:
2562:
2559:
2558:
2557:
2541:
2538:
2537:
2513:
2510:
2509:
2507:
2472:
2467:
2462:
2459:
2458:
2456:
2437:
2434:
2433:
2432:
2411:
2403:
2400:
2399:
2374:
2370:
2365:
2341:
2333:
2330:
2329:
2327:
2309:
2306:
2305:
2304:
2286:
2283:
2282:
2281:
2257:
2254:
2253:
2251:
2240:
2218:
2215:
2214:
2192:
2189:
2188:
2164:
2161:
2160:
2158:
2130:
2127:
2126:
2124:
2105:
2101:
2099:
2096:
2095:
2093:
2077:
2074:
2073:
2063:
2061:Generalizations
2055:
2039:
2035:
2033:
2030:
2029:
1989:
1986:
1985:
1958:
1947:
1944:
1943:
1916:
1905:
1902:
1901:
1884:
1880:
1878:
1875:
1874:
1854:
1853:
1831:
1813:
1812:
1807:
1804:
1803:
1800:greedy coloring
1782:
1781:
1759:
1741:
1740:
1735:
1732:
1731:
1715:
1712:
1711:
1685:
1681:
1679:
1676:
1675:
1652:
1648:
1646:
1643:
1642:
1625:
1621:
1619:
1616:
1615:
1593:
1590:
1589:
1563:
1560:
1559:
1557:
1533:
1530:
1529:
1527:
1520:Fulkerson Prize
1495:
1492:
1491:
1489:
1466:
1462:
1460:
1457:
1456:
1431:
1427:
1425:
1422:
1421:
1399:
1396:
1395:
1370:
1366:
1364:
1361:
1360:
1334:
1331:
1330:
1305:
1301:
1299:
1296:
1295:
1271:
1268:
1267:
1265:
1245:
1241:
1239:
1236:
1235:
1205:
1202:
1201:
1184:
1180:
1178:
1175:
1174:
1152:
1149:
1148:
1145:
1123:
1120:
1119:
1117:
1089:
1086:
1085:
1045:
1042:
1041:
1023:
1020:
1019:
1017:
996:
993:
992:
990:
974:
971:
970:
953:
949:
947:
944:
943:
927:
924:
923:
907:
904:
903:
878:
875:
874:
872:Hadwiger number
850:
846:
844:
841:
840:
838:
817:
811:
810:
809:
807:
804:
803:
769:
766:
765:
764:
747:
741:
740:
739:
737:
734:
733:
716:
710:
709:
708:
706:
703:
702:
680:
676:
674:
671:
670:
668:
647:
644:
643:
641:
622:
618:
616:
613:
612:
610:
592:
589:
588:
586:
568:
565:
564:
563:
535:
532:
531:
515:
512:
511:
492:
488:
486:
483:
482:
480:
464:
461:
460:
449:
412:
409:
408:
406:
387:
384:
383:
366:
362:
360:
357:
356:
333:
330:
329:
304:
301:
300:
284:
281:
280:
264:
261:
260:
216:
213:
212:
211:
178:
175:
174:
173:
149:
145:
143:
140:
139:
119:
116:
115:
114:states that if
92:
91:
86:
61:
58:
57:
56:
40:
37:
36:
30:
23:
17:
12:
11:
5:
4141:
4131:
4130:
4125:
4120:
4115:
4113:Graph coloring
4100:
4099:
4080:(4): 482–492,
4069:
4038:
4003:
3984:(1): 180–181,
3968:
3943:(2): 261–265,
3932:
3907:(3): 279–361,
3888:
3867:
3865:(9): 1084–1086
3844:
3794:
3790:
3777:
3760:(4): 655–659,
3742:
3717:(4): 307–316,
3706:
3689:(3): 327–353,
3677:
3668:
3657:
3638:(4): 728–731,
3624:
3613:
3601:Hadwiger, Hugo
3597:
3562:
3545:(2): 141–143,
3525:
3512:
3499:
3481:
3470:(2): 268–274,
3459:
3435:
3424:(3): 195–199,
3397:
3356:Wood, David R.
3350:
3348:
3345:
3343:
3342:
3330:
3314:
3302:
3286:
3274:
3259:
3247:
3235:
3231:big O notation
3218:
3199:
3184:
3181:
3178:
3174:
3151:
3147:
3116:
3113:
3110:
3106:
3083:
3079:
3062:
3050:
3033:
3029:Diestel (2017)
3020:
3018:
3015:
3007:Petersen graph
2978:
2974:
2951:
2925:
2918:
2915:
2912:
2907:
2902:
2897:
2894:
2891:
2888:
2885:
2882:
2859:
2855:
2828:
2824:
2803:
2783:
2757:
2753:
2732:
2712:
2688:
2685:
2682:
2679:
2657:
2654:
2651:
2648:
2626:
2623:
2620:
2594:
2590:
2566:
2545:
2523:
2520:
2517:
2482:
2479:
2476:
2470:
2466:
2441:
2420:
2415:
2410:
2407:
2385:
2382:
2377:
2373:
2368:
2364:
2361:
2358:
2355:
2349:
2346:
2340:
2337:
2313:
2290:
2267:
2264:
2261:
2228:
2225:
2222:
2202:
2199:
2196:
2174:
2171:
2168:
2140:
2137:
2134:
2108:
2104:
2081:
2062:
2059:
2042:
2038:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1973:
1968:
1965:
1962:
1957:
1954:
1951:
1931:
1926:
1923:
1920:
1915:
1912:
1909:
1887:
1883:
1857:
1850:
1847:
1844:
1841:
1838:
1835:
1830:
1827:
1824:
1821:
1816:
1811:
1785:
1778:
1775:
1772:
1769:
1766:
1763:
1758:
1755:
1752:
1749:
1744:
1739:
1719:
1694:
1691:
1688:
1684:
1661:
1658:
1655:
1651:
1628:
1624:
1603:
1600:
1597:
1573:
1570:
1567:
1543:
1540:
1537:
1505:
1502:
1499:
1469:
1465:
1434:
1430:
1409:
1406:
1403:
1373:
1369:
1344:
1341:
1338:
1321:minor are the
1308:
1304:
1281:
1278:
1275:
1248:
1244:
1215:
1212:
1209:
1187:
1183:
1162:
1159:
1156:
1144:
1141:
1127:
1102:
1099:
1096:
1093:
1073:
1070:
1067:
1064:
1061:
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1055:
1052:
1049:
1027:
1000:
978:
956:
952:
931:
911:
891:
888:
885:
882:
853:
849:
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814:
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776:
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750:
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683:
679:
651:
625:
621:
596:
572:
545:
542:
539:
519:
495:
491:
468:
453:contrapositive
448:
445:
435:, was made by
416:
402:vertices as a
391:
369:
365:
354:complete graph
337:
308:
288:
268:
232:
229:
226:
223:
220:
197:
194:
191:
188:
185:
182:
152:
148:
123:
87:
79:complete graph
65:
44:
31:
25:
15:
9:
6:
4:
3:
2:
4140:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4110:
4108:
4097:
4093:
4088:
4083:
4079:
4075:
4070:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4042:Wagner, Klaus
4039:
4036:
4032:
4027:
4022:
4018:
4014:
4013:
4008:
4007:Voigt, Margit
4004:
4001:
3997:
3992:
3987:
3983:
3979:
3978:
3973:
3969:
3966:
3962:
3958:
3954:
3950:
3946:
3942:
3938:
3933:
3930:
3926:
3922:
3918:
3914:
3910:
3906:
3902:
3901:
3900:Combinatorica
3893:
3892:-free graphs"
3887:
3880:
3879:Thomas, Robin
3876:
3875:Seymour, Paul
3872:
3868:
3864:
3860:
3853:
3849:
3845:
3842:
3838:
3834:
3830:
3825:
3820:
3816:
3812:
3811:
3792:
3788:
3778:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3750:Thomas, Robin
3747:
3743:
3740:
3736:
3732:
3728:
3724:
3720:
3716:
3712:
3711:Combinatorica
3707:
3704:
3700:
3696:
3692:
3688:
3684:
3683:Combinatorica
3681:as a minor",
3676:
3667:
3662:
3658:
3655:
3651:
3646:
3641:
3637:
3633:
3627:
3623:
3618:
3614:
3610:
3606:
3602:
3598:
3595:
3591:
3587:
3583:
3579:
3575:
3574:Seymour, Paul
3571:
3567:
3563:
3560:
3556:
3552:
3548:
3544:
3540:
3539:
3538:Combinatorica
3534:
3530:
3526:
3523:
3519:
3515:
3509:
3505:
3500:
3496:
3491:
3487:
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3295:
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3283:
3278:
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3251:
3244:
3239:
3232:
3216:
3209:. The letter
3208:
3203:
3182:
3179:
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3145:
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3054:
3047:
3042:
3040:
3038:
3030:
3025:
3021:
3014:
3012:
3008:
3004:
3003:edge coloring
3000:
2997:, that every
2996:
2995:snark theorem
2976:
2972:
2949:
2940:
2916:
2913:
2910:
2905:
2895:
2892:
2886:
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2857:
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2621:
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2505:
2504:list coloring
2501:
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2356:
2353:
2347:
2344:
2335:
2311:
2288:
2265:
2262:
2259:
2249:
2248:random graphs
2245:
2226:
2223:
2220:
2200:
2197:
2194:
2172:
2169:
2166:
2156:
2155:Catlin (1979)
2138:
2135:
2132:
2106:
2102:
2079:
2071:
2067:
2058:
2040:
2036:
2012:
2009:
2006:
2003:
2000:
1997:
1991:
1966:
1963:
1960:
1955:
1949:
1924:
1921:
1918:
1913:
1907:
1885:
1881:
1871:
1845:
1839:
1836:
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1767:
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1453:Möbius ladder
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1371:
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458:
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438:
437:Hugo Hadwiger
434:
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414:
405:
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363:
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324:
321:
306:
286:
266:
259:
255:
250:
248:
230:
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224:
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195:
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4016:
4010:
3981:
3980:, Series B,
3975:
3940:
3936:
3904:
3898:
3885:
3862:
3858:
3848:Pegg, Ed Jr.
3814:
3808:
3757:
3753:
3714:
3710:
3686:
3682:
3674:
3665:
3635:
3634:, Series B,
3631:
3625:
3621:
3608:
3604:
3588:(1): 20–29,
3585:
3581:
3542:
3536:
3504:Graph Theory
3503:
3485:
3467:
3463:
3445:
3439:
3421:
3415:
3401:Bollobás, B.
3386:10.37236/719
3367:
3363:
3333:
3317:
3305:
3294:Voigt (1993)
3289:
3277:
3262:
3250:
3238:
3202:
3065:
3053:
3024:
2941:
2771:
2702:
2498:
2070:subdivisions
2066:György Hajós
2064:
1872:
1710:Every graph
1709:
1674:minor and a
1587:
1484:
1392:Klaus Wagner
1355:implies the
1328:
1263:
1146:
1113:denotes the
1014:
922:is the size
869:
700:
558:
450:
430:
251:
111:
108:graph theory
105:
99:
4123:Conjectures
4052:: 570–590,
3570:Reed, Bruce
3566:Geelen, Jim
3529:Erdős, Paul
3405:Erdős, Paul
3370:(1): P232,
3338:Pegg (2002)
3011:W. T. Tutte
2999:cubic graph
2607:-minor-free
1449:clique-sums
902:of a graph
138:and has no
4107:Categories
3824:1910.09378
3630:-minors",
3495:2108.01633
3347:References
2187:the cases
172:satisfies
19:See also:
4066:123534907
3965:124801301
3611:: 133–143
3377:1110.2272
2914:
2881:χ
2772:odd minor
2622:≥
2519:≤
2478:
2381:
2357:ε
2354:−
2336:≥
2260:ε
2170:≥
2136:≤
2010:
2004:
1964:
1922:
1837:
1765:
1539:≤
1147:The case
1092:χ
1060:≤
1048:χ
796:-colored,
778:−
583:-coloring
541:−
326:subgraphs
323:connected
228:≤
222:≤
181:χ
168:then its
3881:(1993),
3850:(2002),
3807:minor",
3739:15736799
3703:41451753
3407:(1980),
3394:13822279
3358:(2011),
3005:has the
1870:colors.
556:colors.
320:disjoint
136:loopless
4096:2232386
4035:1235909
4000:1290638
3957:0735367
3929:9608738
3921:1238823
3841:4576840
3774:0831801
3731:0779891
3654:2518204
3559:1266711
3522:3822066
2577:-vertex
2452:-vertex
2328:number
2324:-vertex
2239:remain
2056:minors.
1707:minor.
1261:minor.
839:minor,
763:can be
585:of any
561:minimal
76:-vertex
55:have a
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3392:
3197:minor.
3133:, and
2843:sparse
2770:as an
2611:every
2556:has a
2250:: for
2094:graph
1084:where
669:graph
611:graph
587:graph
481:graph
256:of an
110:, the
4062:S2CID
3961:S2CID
3925:S2CID
3895:(PDF)
3855:(PDF)
3819:arXiv
3735:S2CID
3699:S2CID
3555:S2CID
3490:arXiv
3412:(PDF)
3390:S2CID
3372:arXiv
3017:Notes
2964:than
2241:open.
1232:cycle
802:that
559:In a
510:then
404:minor
166:minor
83:minor
81:as a
3508:ISBN
2508:For
2263:>
2252:any
2213:and
2159:for
2153:but
2125:for
1588:For
1569:>
1558:all
1528:for
1490:for
1266:for
870:The
350:edge
279:use
193:<
4082:doi
4054:doi
4050:114
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3986:doi
3945:doi
3909:doi
3829:doi
3815:422
3762:hdl
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3691:doi
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3672:or
3640:doi
3590:doi
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3426:doi
3382:doi
3164:or
3096:or
2911:log
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2475:log
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1961:log
1919:log
1834:log
1762:log
1118:of
1018:of
991:of
701:If
642:of
407:of
382:on
328:of
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4109::
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4084::
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