398:
267:
102:
497:
138:
324:
154:
30:
449:
598:
Zang, Wenan; Jing, Guangming; Chen, Guantao (2019-01-29). "Proof of the
Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs".
419:
423:
393:{\displaystyle \operatorname {\chi '} G\geq \max(\operatorname {\Delta } G,\lceil \operatorname {\Gamma } G\rceil ).}
262:{\displaystyle \operatorname {\Gamma } G=\max _{H\subset G}{\frac {|E(H)|}{\lfloor {\frac {1}{2}}|V(H)|\rfloor }}.}
442:
was announced by Chen, Jing, and Zang in the paper . Part of their proof was to find a suitable generalization of
110:
675:
621:
504:
97:{\displaystyle \operatorname {\chi '} G\leq \max(1+\operatorname {\Delta } G,\,\operatorname {\Gamma } G)}
680:
427:
521:
443:
300:
649:
599:
439:
630:
500:
576:
526:
516:
404:
669:
141:
408:
407:. It is hard to find other examples. It is currently unknown whether there are any
312:
17:
546:
415:
296:
273:
619:
Goldberg, Mark (1984). "Edge-coloring of multigraphs: Recoloring technique".
634:
492:{\displaystyle \operatorname {\chi '} G\leq 1+\operatorname {\Delta } G}
604:
577:"The Goldberg-Seymour Conjecture on the edge coloring of multigraphs"
654:
648:
Jing, Guangming (2023-08-29). "On Edge
Coloring of Multigraphs".
499:) to multigraphs. In 2023, Jing announced a new proof with a
403:
When does equality not hold? It does not hold for the
452:
327:
157:
113:
33:
503:edge coloring algorithm achieving the conjectured
491:
392:
261:
132:
96:
347:
173:
53:
430:, who arrived to it independently of Goldberg.
8:
547:"Problems in Graph Theory and Combinatorics"
381:
367:
250:
215:
653:
603:
478:
453:
451:
370:
353:
328:
326:
245:
228:
218:
208:
191:
188:
176:
158:
156:
114:
112:
80:
79:
65:
34:
32:
133:{\displaystyle \operatorname {\chi '} G}
538:
272:Note this above quantity is twice the
7:
593:
591:
589:
570:
568:
566:
446:(which says that for simple graphs
418:is named after Mark K. Goldberg of
479:
411:for which equality does not hold.
371:
354:
159:
81:
66:
14:
420:Rensselaer Polytechnic Institute
318:(but can have parallel edges):
384:
350:
246:
242:
236:
229:
209:
205:
199:
192:
91:
56:
1:
311:It is already known that for
280:. It is sometimes called the
582:. Georgia State University.
22:Goldberg–Seymour conjecture
697:
551:faculty.math.illinois.edu
575:Jing, Guangming (2018).
622:Journal of Graph Theory
517:Petersen graph#Coloring
635:10.1002/jgt.3190080115
493:
394:
263:
134:
98:
494:
395:
264:
142:edge chromatic number
135:
99:
450:
438:In 2019, an alleged
428:Princeton University
325:
155:
111:
31:
522:Fractional coloring
489:
390:
259:
187:
130:
94:
254:
226:
172:
688:
660:
659:
657:
645:
639:
638:
616:
610:
609:
607:
595:
584:
583:
581:
572:
561:
560:
558:
557:
543:
498:
496:
495:
490:
482:
462:
461:
444:Vizing's theorem
399:
397:
396:
391:
374:
357:
337:
336:
268:
266:
265:
260:
255:
253:
249:
232:
227:
219:
213:
212:
195:
189:
186:
162:
139:
137:
136:
131:
123:
122:
103:
101:
100:
95:
84:
69:
43:
42:
696:
695:
691:
690:
689:
687:
686:
685:
666:
665:
664:
663:
647:
646:
642:
618:
617:
613:
597:
596:
587:
579:
574:
573:
564:
555:
553:
545:
544:
540:
535:
513:
501:polynomial-time
454:
448:
447:
436:
434:Announced proof
329:
323:
322:
309:
214:
190:
153:
152:
115:
109:
108:
35:
29:
28:
12:
11:
5:
694:
692:
684:
683:
678:
676:Graph coloring
668:
667:
662:
661:
640:
629:(1): 123–137.
611:
585:
562:
537:
536:
534:
531:
530:
529:
527:Graph coloring
524:
519:
512:
509:
488:
485:
481:
477:
474:
471:
468:
465:
460:
457:
435:
432:
405:Petersen graph
401:
400:
389:
386:
383:
380:
377:
373:
369:
366:
363:
360:
356:
352:
349:
346:
343:
340:
335:
332:
308:
305:
270:
269:
258:
252:
248:
244:
241:
238:
235:
231:
225:
222:
217:
211:
207:
204:
201:
198:
194:
185:
182:
179:
175:
171:
168:
165:
161:
129:
126:
121:
118:
105:
104:
93:
90:
87:
83:
78:
75:
72:
68:
64:
61:
58:
55:
52:
49:
46:
41:
38:
13:
10:
9:
6:
4:
3:
2:
693:
682:
679:
677:
674:
673:
671:
656:
651:
644:
641:
636:
632:
628:
624:
623:
615:
612:
606:
601:
594:
592:
590:
586:
578:
571:
569:
567:
563:
552:
548:
542:
539:
532:
528:
525:
523:
520:
518:
515:
514:
510:
508:
506:
502:
486:
483:
475:
472:
469:
466:
463:
458:
455:
445:
441:
433:
431:
429:
425:
421:
417:
412:
410:
409:planar graphs
406:
387:
378:
375:
364:
361:
358:
344:
341:
338:
333:
330:
321:
320:
319:
317:
314:
306:
304:
302:
298:
294:
289:
287:
283:
279:
275:
256:
239:
233:
223:
220:
202:
196:
183:
180:
177:
169:
166:
163:
151:
150:
149:
147:
143:
127:
124:
119:
116:
88:
85:
76:
73:
70:
62:
59:
50:
47:
44:
39:
36:
27:
26:
25:
23:
19:
643:
626:
620:
614:
605:1901.10316v1
554:. Retrieved
550:
541:
437:
424:Paul Seymour
413:
402:
315:
310:
292:
290:
285:
281:
277:
271:
145:
106:
24:states that
21:
18:graph theory
15:
681:Conjectures
670:Categories
655:2308.15588
556:2019-05-05
533:References
416:conjecture
307:Background
299:(can have
297:multigraph
274:arboricity
484:
480:Δ
470:≤
464:
456:χ
382:⌉
376:
372:Γ
368:⌈
359:
355:Δ
345:≥
339:
331:χ
295:can be a
251:⌋
216:⌊
181:⊂
164:
160:Γ
125:
117:χ
86:
82:Γ
71:
67:Δ
51:≤
45:
37:χ
511:See also
459:′
334:′
313:loopless
284:of
276:of
120:′
40:′
282:density
140:is the
291:Above
107:where
20:, the
650:arXiv
600:arXiv
580:(PDF)
505:bound
440:proof
414:This
301:loops
422:and
148:and
631:doi
426:of
348:max
303:).
174:max
144:of
54:max
16:In
672::
625:.
588:^
565:^
549:.
507:.
288:.
658:.
652::
637:.
633::
627:8
608:.
602::
559:.
487:G
476:+
473:1
467:G
388:.
385:)
379:G
365:,
362:G
351:(
342:G
316:G
293:G
286:G
278:G
257:.
247:|
243:)
240:H
237:(
234:V
230:|
224:2
221:1
210:|
206:)
203:H
200:(
197:E
193:|
184:G
178:H
170:=
167:G
146:G
128:G
92:)
89:G
77:,
74:G
63:+
60:1
57:(
48:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.