28:
1555:
1326:
1088:
3772:
A team of UT Dallas computer science students and their faculty adviser have improved upon a longstanding solution to a mathematical conundrum known as the pancake problem. The previous best solution, which stood for almost 30 years, was devised by Bill Gates and one of his
Harvard instructors,
2810:
was the operation connecting two permutations. If we allow non-prefixed reversals (as if we were flipping with two spatulas instead of one) then four classes of pancake graphs can be defined. Every pancake graph embeds in all higher-order pancake graphs of the same family.
3360:), much attention is paid to them as a model of interconnection networks for parallel computers. When we regard the pancake graphs as the model of the interconnection networks, the diameter of the graph is a measure that represents the delay of communication.
678:
2364:
In 2011, Laurent
Bulteau, Guillaume Fertin, and Irena Rusu proved that the problem of finding the shortest sequence of flips for a given stack of pancakes is NP-hard, thereby answering a question that had been open for over three decades.
1363:
1131:
905:
3363:
Pancake flipping has biological applications as well, in the field of genetic examinations. In one kind of large-scale mutations, a large segment of the genome gets reversed, which is analogous to the burnt pancake problem.
3381:
Asai, Shogo; Kounoike, Yuusuke; Shinano, Yuji; Kaneko, Keiichi (2006). "Computing the diameter of 17-pancake graph using a PC cluster". In Nagel, Wolfgang E.; Walter, Wolfgang V.; Lehner, Wolfgang (eds.).
2021:
2063:
The pancake sorting problem (obtaining the pancake number) and obtaining the diameter of the pancake graph are equivalents. One of the main difficulties in solving this problem is the complicated
409:
136:
2796:
529:
2561:
562:
816:
1779:
758:
3311:
2053:
1831:
1123:
3263:
3145:
3048:
2468:
1926:
897:
1885:
1671:
862:
2696:
2428:
3756:
3700:
1355:
3223:
3105:
2981:
2870:
2587:
3876:
Akl, S.G.; Qiu, K.; Stojmenović, I. (1993). "Fundamental algorithms for the star and pancake interconnection networks with applications to computational geometry".
2186:
2924:
3008:
2897:
1550:{\displaystyle \chi (P_{n})\leq {\begin{cases}n-(k+4),&{\text{if }}n\equiv k{\pmod {4}},k=1,2,3;\\n-8,&{\text{if }}n\equiv k{\pmod {4}},k=0.\end{cases}}}
1321:{\displaystyle \chi (P_{n})\leq {\begin{cases}n-(k+2),&{\text{if }}n\equiv k{\pmod {4}},k=1,3;\\n-4,&{\text{if }}n\equiv k{\pmod {4}},k=0,2;\end{cases}}}
344:
71:
3332:
3187:
3166:
3069:
2945:
2627:
2488:
2105:
1590:
1083:{\displaystyle \chi (P_{n})\leq {\begin{cases}n-k,&{\text{if }}n\equiv k{\pmod {4}},k=1,3;\\n-2,&{\text{if }}n\equiv k{\pmod {4}},k=0,2;\end{cases}}}
2602:
2080:
299:
is the minimum number of flips required for a given number of pancakes. Obtaining the pancake number is equivalent to the problem of obtaining the
4011:
3986:
Kumar, V., Grama, A., Gupta, A., Karypis, G.: Introduction to
Parallel Computing: Design and Analysis of Algorithms. Benjamin/Cummings (1994)
3399:
238:
3643:
3384:
Euro-Par 2006, Parallel
Processing, 12th International Euro-Par Conference, Dresden, Germany, August 28 – September 1, 2006, Proceedings
3913:
Bass, D.W.; Sudborough, I.H. (March 2003). "Pancake problems with restricted prefix reversals and some corresponding Cayley networks".
2377:, the bottom of each pancake in the pile is burnt, and the sort must be completed with the burnt side of every pancake down. It is a
3684:
1947:
3760:
3727:
3950:
Berthomé, P.; Ferreira, A.; Perennes, S. (December 1996). "Optimal information dissemination in star and pancake networks".
2354:
357:
84:
3527:
Blanco, Saúl; Buehrle, Charles (June 20, 2023). "Bounds on the genus for 2-cell embeddings of prefix-reversal graphs".
459:
455:
216:
212:
208:
2745:
475:
3785:
Chitturi, B.; Fahle, W.; Meng, Z.; Morales, L.; Shields, C. O.; Sudborough, I. H.; Voit, W. (August 31, 2009).
3793:. Graphs, Games and Computation: Dedicated to Professor Burkhard Monien on the Occasion of his 65th Birthday.
2504:
673:{\displaystyle n!\left({\frac {n-4}{6}}\right)+1\leq \gamma (P_{n})\leq n!\left({\frac {3n-10}{12}}\right)+1.}
3349:
300:
204:
3344:
Since pancake graphs have many interesting properties such as symmetric and recursive structures (they are
3959:
3922:
3885:
3718:
2308:
766:
1749:
714:
46:
3269:
2064:
2026:
1804:
1739:
466:
142:
3890:
2358:
1394:
1162:
1096:
936:
3964:
3927:
3863:
3232:
3114:
3014:
2437:
1890:
870:
77:
1852:
1640:
831:
3827:
3694:
3528:
3442:
3424:
3415:
Deng, Yun-Ping; Xiao-Dong, Zhang (March 31, 2012). "Automorphism Groups of the
Pancake Graphs".
3757:"Team Bests Young Bill Gates With Improved Answer to So-Called Pancake Problem in Mathematics"
3680:
3395:
1334:
351:
196:
3969:
3932:
3895:
3837:
3798:
3736:
3655:
3622:
3564:
3478:
3434:
3387:
3198:
3080:
2956:
2845:
2671:
2566:
2403:
1782:
156:
2164:
3585:
3357:
544:
284:
179:
166:
2903:
3995:
Quinn, M.J.: Parallel
Computing: Theory and Practice, second edition. McGraw Hill (1994)
2990:
2879:
326:
53:
3317:
3172:
3151:
3054:
2930:
2612:
2494:
2473:
2244:
The pancake number, which is the minimum number of flips required to sort any stack of
2090:
1575:
709:
689:
3936:
4005:
3818:
Bulteau, Laurent; Fertin, Guillaume; Rusu, Irena (2015). "Pancake
Flipping Is Hard".
3741:
3722:
2431:
761:
320:
295:
can be inserted at any point in the stack and used to flip all pancakes above it. A
287:
is the colloquial term for the mathematical problem of sorting a disordered stack of
192:
3446:
1801:≥ 4) pancake graph: every vertex belongs to exactly one 6-cycle. The graph contains
40:
by assigning a different element from the set {1, 2, 3, 4} as a suffix to each copy.
3386:. Lecture Notes in Computer Science. Vol. 4128. Springer. pp. 1114–1124.
3345:
251:
200:
3502:
3675:
Fertin, G. and
Labarre, A. and Rusu, I. and Tannier, E. and Vialette, S. (2009).
3353:
2498:
247:
3841:
3759:. University of Texas at Dallas News Center. September 17, 2008. Archived from
3627:
3610:
27:
3803:
3786:
3714:
3660:
3483:
3466:
3438:
2304:
1560:
The following table discusses specific chromatic number values for some small
821:
There are effective algorithms for the proper (n−1)-coloring and total
281:
and its edges are given between permutations transitive by prefix reversals.
2806:
Both in the original pancake sorting problem and the burnt pancake problem,
1781:(but there are no cycles of length 3, 4 or 5). It implies that the graph is
3899:
3391:
3773:
Christos
Papadimitriou, several years before Microsoft was established.
292:
288:
3973:
3569:
3552:
3593:
The 23rd
Workshop on Combinatorial Mathematics and Computation Theory
3533:
2023:
different 8-cycles; a maximal set of independent 8-cycles contains
3832:
3429:
2493:
For its variant David S. Cohen (best known by his pen name "
2597:
2075:
1543:
1314:
1076:
3352:), sublogarithmic degree and diameter, and are relatively
1785:
and any two vertices can be joined by a Hamiltonian path.
430:, by assigning a different element from the set {1, 2, …,
3787:"An (18/11)n upper bound for sorting by prefix reversals"
3510:
The International Arab Journal of Information Technology
2016:{\displaystyle {\frac {n!(n^{3}+12n^{2}-103n+176)}{16}}}
3609:
Konstantinova, E.V.; Medvedev, A.N. (April 26, 2013).
3952:
IEEE Transactions on Parallel and Distributed Systems
3642:
Konstantinova, E.V.; Medvedev, A.N. (April 1, 2010).
3586:"Cycle embedding in pancake interconnection networks"
3320:
3272:
3235:
3201:
3175:
3154:
3117:
3083:
3057:
3017:
2993:
2959:
2933:
2906:
2882:
2848:
2748:
2674:
2615:
2569:
2507:
2476:
2440:
2406:
2167:
2093:
2029:
1950:
1893:
1855:
1807:
1752:
1643:
1578:
1568:
Specific, or probable values of the chromatic number
1366:
1337:
1134:
1099:
908:
873:
834:
769:
717:
565:
478:
362:
360:
329:
89:
87:
56:
222:
188:
178:
165:
155:
141:
76:
45:
20:
3326:
3305:
3257:
3217:
3181:
3160:
3139:
3099:
3063:
3042:
3002:
2975:
2939:
2918:
2891:
2864:
2790:
2690:
2621:
2581:
2555:
2482:
2462:
2422:
2180:
2099:
2047:
2015:
1920:
1879:
1825:
1773:
1665:
1584:
1549:
1349:
1320:
1117:
1082:
891:
856:
810:
752:
672:
523:
404:{\displaystyle {\dfrac {1}{2}}~n!\left(n-1\right)}
403:
338:
273:is a graph whose vertices are the permutations of
131:{\displaystyle {\dfrac {1}{2}}~n!\left(n-1\right)}
130:
65:
3501:Nguyen, Quan; Bettayeb, Said (November 5, 2009).
3297:
3276:
3034:
3021:
864:chromatic number the following limits are known:
36:can be constructed recursively from 4 copies of P
2300:,) but the exact value remains an open problem.
1738:≥ 3) pancake graph there is always at least one
420:can be constructed recursively from n copies of
2791:{\displaystyle g(P_{n})=8{\text{, if }}n>2.}
524:{\displaystyle g(P_{n})=6,{\text{ if }}n>2.}
350: − 1, hence, according to the
3915:Journal of Parallel and Distributed Computing
3644:"Cycles of length seven in the pancake graph"
2397:is the graph representation of this problem.
8:
3699:: CS1 maint: multiple names: authors list (
3553:"Chromatic Properties of the Pancake Graphs"
3227:The burnt pancake problem with two spatulas
2332:. This was improved, thirty years later, to
1849:≥ 4) pancake graph: every vertex belongs to
3546:
3544:
3465:Compeau, Phillip E.C. (September 6, 2011).
3963:
3926:
3889:
3856:On the problem of sorting burnt pancakes.
3831:
3802:
3740:
3659:
3626:
3584:Sheu, Jyh-Jian; Tan, Jimmy J. M. (2006).
3568:
3532:
3496:
3494:
3482:
3428:
3319:
3296:
3275:
3273:
3271:
3240:
3234:
3209:
3200:
3174:
3153:
3122:
3116:
3091:
3082:
3056:
3033:
3020:
3018:
3016:
2992:
2967:
2958:
2932:
2905:
2881:
2856:
2847:
2774:
2759:
2747:
2679:
2673:
2614:
2568:
2529:
2514:
2506:
2475:
2445:
2439:
2411:
2405:
2172:
2166:
2092:
2030:
2028:
1983:
1967:
1951:
1949:
1892:
1854:
1808:
1806:
1751:
1654:
1642:
1577:
1512:
1498:
1437:
1423:
1389:
1377:
1365:
1336:
1274:
1260:
1205:
1191:
1157:
1145:
1133:
1098:
1036:
1022:
967:
953:
931:
919:
907:
872:
845:
833:
787:
774:
768:
735:
722:
716:
639:
617:
576:
564:
507:
489:
477:
361:
359:
328:
88:
86:
55:
3604:
3602:
2813:
2591:
2556:{\displaystyle 3n/2\leq P'_{n}\leq 2n-2}
2069:
1833:independent (vertex-disjoint) 6-cycles.
1566:
3820:Journal of Computer and System Sciences
3723:"Bounds for Sorting by Prefix Reversal"
3551:Konstantinova, Elena (August 1, 2017).
3460:
3458:
3456:
3373:
2985:The original problem with two spatulas
2739:The girth of a burnt pancake graph is:
2250:pancakes has been shown to lie between
1944:≥ 4) pancake graph: the graph contains
3692:
3677:Combinatorics of Genome Rearrangements
3557:Discussiones Mathematicae Graph Theory
2563:(when the upper limit is only true if
2385:is "burnt side up" a negative element
17:
2874:The original pancake sorting problem
7:
811:{\displaystyle \chi _{e}(P_{n})=n-1}
3611:"Small cycles in the Pancake graph"
1520:
1513:
1445:
1438:
1282:
1275:
1213:
1206:
1044:
1037:
975:
968:
3280:
3025:
1774:{\displaystyle 6\leq \ell \leq n!}
753:{\displaystyle \chi _{t}(P_{n})=n}
14:
3503:"On the Genus of Pancake Network"
3864:DOI:10.1016/0166-218X(94)00009-3
2353:by a team of researchers at the
2067:structure of the pancake graph.
556:is bounded below and above by:
26:
3306:{\displaystyle {n+1 \choose 2}}
2840:unsigned prefix reversal graph
2802:Other classes of pancake graphs
2048:{\displaystyle {\frac {n!}{8}}}
1826:{\displaystyle {\frac {n!}{6}}}
306:The pancake graph of dimension
3417:Information Processing Letters
2765:
2752:
2004:
1960:
1915:
1903:
1874:
1862:
1660:
1647:
1524:
1514:
1449:
1439:
1415:
1403:
1383:
1370:
1286:
1276:
1217:
1207:
1183:
1171:
1151:
1138:
1118:{\displaystyle 9\leq n\leq 16}
1048:
1038:
979:
969:
925:
912:
851:
838:
793:
780:
741:
728:
692:properties of pancake graphs.
623:
610:
495:
482:
239:Table of graphs and parameters
1:
4012:Parametric families of graphs
3937:10.1016/S0743-7315(03)00033-9
3862:61, Nr. 2, 1995, S. 105–120.
3860:Discrete Applied Mathematics.
3854:David S. Cohen, Manuel Blum:
3615:Ars Mathematica Contemporanea
3258:{\displaystyle 2^{n}\cdot n!}
3140:{\displaystyle 2^{n}\cdot n!}
3075:signed prefix reversal graph
3043:{\displaystyle {n \choose 2}}
2463:{\displaystyle 2^{n}\cdot n!}
2355:University of Texas at Dallas
1921:{\displaystyle n!\cdot (n-3)}
1887:7-cycles. The graph contains
892:{\displaystyle 4\leq n\leq 8}
825:-coloring of pancake graphs.
3791:Theoretical Computer Science
3742:10.1016/0012-365X(79)90068-2
3648:Diskretn. Anal. Issled. Oper
3471:Discrete Applied Mathematics
2357:, led by Founders Professor
1880:{\displaystyle 7\cdot (n-3)}
1666:{\displaystyle \chi (P_{n})}
857:{\displaystyle \chi (P_{n})}
434:} as a suffix to each copy.
4028:
3842:10.1016/j.jcss.2015.02.003
3628:10.26493/1855-3974.214.0e8
3109:The burnt pancake problem
2815:Classes of pancake graphs
2373:In a variation called the
1931:About the 8-cycles of the
1836:About the 7-cycles of the
1788:About the 6-cycles of the
3804:10.1016/j.tcs.2008.04.045
3661:10.1016/j.tcs.2008.04.045
3484:10.1016/j.dam.2011.06.013
3467:"Girth of pancake graphs"
3439:10.1016/j.ipl.2011.12.010
2361:(Chitturi et al., 2009).
237:
25:
2951:unsigned reversal graph
2393:in the permutation. The
1350:{\displaystyle n\geq 17}
346:vertices. Its degree is
291:in order of size when a
2430:burnt pancake graph is
2311:gave an upper bound of
708:≥ 3) pancake graph has
3900:10.1002/net.3230230403
3328:
3307:
3259:
3219:
3218:{\displaystyle SR_{n}}
3193:signed reversal graph
3183:
3162:
3141:
3101:
3100:{\displaystyle SP_{n}}
3065:
3044:
3004:
2977:
2976:{\displaystyle UR_{n}}
2941:
2920:
2893:
2866:
2865:{\displaystyle UP_{n}}
2792:
2692:
2691:{\displaystyle P'_{n}}
2623:
2583:
2582:{\displaystyle n>9}
2557:
2484:
2464:
2424:
2423:{\displaystyle P'_{n}}
2309:Christos Papadimitriou
2182:
2101:
2049:
2017:
1922:
1881:
1827:
1775:
1667:
1586:
1551:
1351:
1322:
1119:
1084:
893:
858:
812:
754:
710:total chromatic number
674:
525:
405:
340:
303:of the pancake graph.
132:
67:
3329:
3308:
3260:
3220:
3184:
3163:
3142:
3102:
3066:
3045:
3005:
2978:
2942:
2921:
2894:
2867:
2793:
2693:
2624:
2593:Burnt pancake numbers
2584:
2558:
2501:proved in 1995, that
2485:
2465:
2425:
2375:burnt pancake problem
2183:
2181:{\displaystyle P_{n}}
2102:
2050:
2018:
1923:
1882:
1828:
1776:
1668:
1587:
1552:
1352:
1323:
1120:
1085:
894:
859:
813:
755:
688:There are some known
675:
526:
406:
341:
133:
68:
3728:Discrete Mathematics
3392:10.1007/11823285_117
3318:
3270:
3233:
3199:
3173:
3152:
3115:
3081:
3055:
3015:
2991:
2957:
2931:
2904:
2880:
2846:
2746:
2672:
2613:
2567:
2505:
2474:
2438:
2404:
2165:
2091:
2027:
1948:
1928:different 7-cycles.
1891:
1853:
1805:
1750:
1641:
1576:
1364:
1335:
1132:
1097:
906:
871:
832:
767:
715:
684:Chromatic properties
563:
476:
358:
327:
85:
54:
2919:{\displaystyle n-1}
2816:
2687:
2606:
2537:
2419:
2395:burnt pancake graph
2389:is put in place of
2381:, and if a pancake
2369:Burnt pancake graph
2292:(approximately 1.07
2084:
1569:
209:Maximally connected
32:The pancake graph P
3356:(compared to e.g.
3324:
3303:
3255:
3215:
3179:
3158:
3137:
3097:
3061:
3040:
3003:{\displaystyle n!}
3000:
2973:
2937:
2916:
2892:{\displaystyle n!}
2889:
2862:
2835:Girth (if n>2)
2814:
2788:
2688:
2675:
2619:
2592:
2579:
2553:
2525:
2480:
2460:
2420:
2407:
2379:signed permutation
2178:
2097:
2070:
2045:
2013:
1918:
1877:
1823:
1771:
1663:
1582:
1567:
1547:
1542:
1347:
1318:
1313:
1115:
1080:
1075:
889:
854:
808:
750:
670:
521:
401:
371:
339:{\displaystyle n!}
336:
277:symbols from 1 to
184:see in the article
161:see in the article
128:
98:
66:{\displaystyle n!}
63:
3974:10.1109/71.553290
3958:(12): 1292–1300.
3797:(36): 3372–3390.
3719:Papadimitriou, C.
3679:. The MIT Press.
3570:10.7151/dmgt.1978
3477:(15): 1641–1645.
3401:978-3-540-37783-2
3350:vertex-transitive
3337:
3336:
3327:{\displaystyle 4}
3295:
3182:{\displaystyle 8}
3161:{\displaystyle n}
3064:{\displaystyle 4}
3032:
2940:{\displaystyle 6}
2777:
2737:
2736:
2622:{\displaystyle n}
2483:{\displaystyle n}
2242:
2241:
2100:{\displaystyle n}
2043:
2011:
1821:
1721:Cycle enumeration
1718:
1717:
1585:{\displaystyle n}
1501:
1426:
1263:
1194:
1025:
956:
658:
592:
510:
375:
370:
352:handshaking lemma
244:
243:
205:Vertex-transitive
102:
97:
4019:
3996:
3993:
3987:
3984:
3978:
3977:
3967:
3947:
3941:
3940:
3930:
3910:
3904:
3903:
3893:
3873:
3867:
3852:
3846:
3845:
3835:
3826:(8): 1556–1574.
3815:
3809:
3808:
3806:
3782:
3776:
3775:
3769:
3768:
3753:
3747:
3746:
3744:
3711:
3705:
3704:
3698:
3690:
3672:
3666:
3665:
3663:
3639:
3633:
3632:
3630:
3606:
3597:
3596:
3590:
3581:
3575:
3574:
3572:
3548:
3539:
3538:
3536:
3524:
3518:
3517:
3507:
3498:
3489:
3488:
3486:
3462:
3451:
3450:
3432:
3412:
3406:
3405:
3378:
3333:
3331:
3330:
3325:
3312:
3310:
3309:
3304:
3302:
3301:
3300:
3291:
3279:
3264:
3262:
3261:
3256:
3245:
3244:
3224:
3222:
3221:
3216:
3214:
3213:
3188:
3186:
3185:
3180:
3167:
3165:
3164:
3159:
3146:
3144:
3143:
3138:
3127:
3126:
3106:
3104:
3103:
3098:
3096:
3095:
3070:
3068:
3067:
3062:
3049:
3047:
3046:
3041:
3039:
3038:
3037:
3024:
3009:
3007:
3006:
3001:
2982:
2980:
2979:
2974:
2972:
2971:
2946:
2944:
2943:
2938:
2925:
2923:
2922:
2917:
2898:
2896:
2895:
2890:
2871:
2869:
2868:
2863:
2861:
2860:
2817:
2797:
2795:
2794:
2789:
2778:
2775:
2764:
2763:
2697:
2695:
2694:
2689:
2683:
2628:
2626:
2625:
2620:
2607:
2600:
2588:
2586:
2585:
2580:
2562:
2560:
2559:
2554:
2533:
2518:
2489:
2487:
2486:
2481:
2470:, its degree is
2469:
2467:
2466:
2461:
2450:
2449:
2429:
2427:
2426:
2421:
2415:
2352:
2348:
2346:
2345:
2342:
2339:
2331:
2327:
2325:
2324:
2321:
2318:
2291:
2287:
2285:
2284:
2281:
2278:
2270:
2266:
2264:
2263:
2260:
2257:
2249:
2187:
2185:
2184:
2179:
2177:
2176:
2106:
2104:
2103:
2098:
2085:
2078:
2054:
2052:
2051:
2046:
2044:
2039:
2031:
2022:
2020:
2019:
2014:
2012:
2007:
1988:
1987:
1972:
1971:
1952:
1927:
1925:
1924:
1919:
1886:
1884:
1883:
1878:
1832:
1830:
1829:
1824:
1822:
1817:
1809:
1780:
1778:
1777:
1772:
1672:
1670:
1669:
1664:
1659:
1658:
1591:
1589:
1588:
1583:
1570:
1556:
1554:
1553:
1548:
1546:
1545:
1527:
1502:
1499:
1452:
1427:
1424:
1382:
1381:
1356:
1354:
1353:
1348:
1327:
1325:
1324:
1319:
1317:
1316:
1289:
1264:
1261:
1220:
1195:
1192:
1150:
1149:
1124:
1122:
1121:
1116:
1089:
1087:
1086:
1081:
1079:
1078:
1051:
1026:
1023:
982:
957:
954:
924:
923:
898:
896:
895:
890:
863:
861:
860:
855:
850:
849:
817:
815:
814:
809:
792:
791:
779:
778:
759:
757:
756:
751:
740:
739:
727:
726:
679:
677:
676:
671:
663:
659:
654:
640:
622:
621:
597:
593:
588:
577:
530:
528:
527:
522:
511:
508:
494:
493:
410:
408:
407:
402:
400:
396:
373:
372:
363:
345:
343:
342:
337:
157:Chromatic number
137:
135:
134:
129:
127:
123:
100:
99:
90:
72:
70:
69:
64:
30:
18:
4027:
4026:
4022:
4021:
4020:
4018:
4017:
4016:
4002:
4001:
4000:
3999:
3994:
3990:
3985:
3981:
3949:
3948:
3944:
3912:
3911:
3907:
3891:10.1.1.363.4949
3875:
3874:
3870:
3853:
3849:
3817:
3816:
3812:
3784:
3783:
3779:
3766:
3764:
3755:
3754:
3750:
3713:
3712:
3708:
3691:
3687:
3674:
3673:
3669:
3641:
3640:
3636:
3608:
3607:
3600:
3588:
3583:
3582:
3578:
3550:
3549:
3542:
3526:
3525:
3521:
3505:
3500:
3499:
3492:
3464:
3463:
3454:
3414:
3413:
3409:
3402:
3380:
3379:
3375:
3370:
3342:
3316:
3315:
3281:
3274:
3268:
3267:
3236:
3231:
3230:
3205:
3197:
3196:
3171:
3170:
3150:
3149:
3118:
3113:
3112:
3087:
3079:
3078:
3053:
3052:
3019:
3013:
3012:
2989:
2988:
2963:
2955:
2954:
2929:
2928:
2902:
2901:
2878:
2877:
2852:
2844:
2843:
2808:prefix reversal
2804:
2755:
2744:
2743:
2670:
2669:
2611:
2610:
2596:
2594:
2565:
2564:
2503:
2502:
2472:
2471:
2441:
2436:
2435:
2434:, its order is
2402:
2401:
2371:
2343:
2340:
2337:
2336:
2334:
2333:
2322:
2319:
2316:
2315:
2313:
2312:
2282:
2279:
2276:
2275:
2273:
2272:
2261:
2258:
2255:
2254:
2252:
2251:
2245:
2168:
2163:
2162:
2089:
2088:
2074:
2072:
2071:Pancake numbers
2061:
2032:
2025:
2024:
1979:
1963:
1953:
1946:
1945:
1939:
1889:
1888:
1851:
1850:
1844:
1810:
1803:
1802:
1796:
1748:
1747:
1733:
1723:
1650:
1639:
1638:
1574:
1573:
1541:
1540:
1496:
1481:
1480:
1421:
1390:
1373:
1362:
1361:
1333:
1332:
1312:
1311:
1258:
1243:
1242:
1189:
1158:
1141:
1130:
1129:
1095:
1094:
1074:
1073:
1020:
1005:
1004:
951:
932:
915:
904:
903:
869:
868:
841:
830:
829:
783:
770:
765:
764:
762:chromatic index
731:
718:
713:
712:
703:
686:
641:
635:
613:
578:
572:
561:
560:
555:
542:
485:
474:
473:
460:hyper-connected
456:super-connected
449:
440:
429:
419:
386:
382:
356:
355:
325:
324:
318:
285:Pancake sorting
265:
233:
217:Hyper-connected
215:
213:Super-connected
211:
207:
203:
199:
195:
167:Chromatic index
113:
109:
83:
82:
52:
51:
41:
39:
35:
12:
11:
5:
4025:
4023:
4015:
4014:
4004:
4003:
3998:
3997:
3988:
3979:
3965:10.1.1.44.6681
3942:
3928:10.1.1.27.7009
3921:(3): 327–336.
3905:
3884:(4): 215–225.
3868:
3847:
3810:
3777:
3748:
3706:
3685:
3667:
3634:
3598:
3576:
3563:(3): 777–787.
3540:
3519:
3490:
3452:
3423:(7): 264–266.
3407:
3400:
3372:
3371:
3369:
3366:
3341:
3338:
3335:
3334:
3323:
3313:
3299:
3294:
3290:
3287:
3284:
3278:
3265:
3254:
3251:
3248:
3243:
3239:
3228:
3225:
3212:
3208:
3204:
3194:
3190:
3189:
3178:
3168:
3157:
3147:
3136:
3133:
3130:
3125:
3121:
3110:
3107:
3094:
3090:
3086:
3076:
3072:
3071:
3060:
3050:
3036:
3031:
3028:
3023:
3010:
2999:
2996:
2986:
2983:
2970:
2966:
2962:
2952:
2948:
2947:
2936:
2926:
2915:
2912:
2909:
2899:
2888:
2885:
2875:
2872:
2859:
2855:
2851:
2841:
2837:
2836:
2833:
2830:
2827:
2824:
2821:
2803:
2800:
2799:
2798:
2787:
2784:
2781:
2773:
2770:
2767:
2762:
2758:
2754:
2751:
2735:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2686:
2682:
2678:
2666:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2618:
2578:
2575:
2572:
2552:
2549:
2546:
2543:
2540:
2536:
2532:
2528:
2524:
2521:
2517:
2513:
2510:
2495:David X. Cohen
2479:
2459:
2456:
2453:
2448:
2444:
2418:
2414:
2410:
2370:
2367:
2359:Hal Sudborough
2240:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2175:
2171:
2159:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2096:
2060:
2057:
2042:
2038:
2035:
2010:
2006:
2003:
2000:
1997:
1994:
1991:
1986:
1982:
1978:
1975:
1970:
1966:
1962:
1959:
1956:
1935:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1840:
1820:
1816:
1813:
1792:
1770:
1767:
1764:
1761:
1758:
1755:
1729:
1722:
1719:
1716:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1662:
1657:
1653:
1649:
1646:
1635:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1581:
1558:
1557:
1544:
1539:
1536:
1533:
1530:
1526:
1523:
1519:
1516:
1511:
1508:
1505:
1497:
1495:
1492:
1489:
1486:
1483:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1451:
1448:
1444:
1441:
1436:
1433:
1430:
1422:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1395:
1393:
1388:
1385:
1380:
1376:
1372:
1369:
1346:
1343:
1340:
1329:
1328:
1315:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1288:
1285:
1281:
1278:
1273:
1270:
1267:
1259:
1257:
1254:
1251:
1248:
1245:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1219:
1216:
1212:
1209:
1204:
1201:
1198:
1190:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1163:
1161:
1156:
1153:
1148:
1144:
1140:
1137:
1114:
1111:
1108:
1105:
1102:
1091:
1090:
1077:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1050:
1047:
1043:
1040:
1035:
1032:
1029:
1021:
1019:
1016:
1013:
1010:
1007:
1006:
1003:
1000:
997:
994:
991:
988:
985:
981:
978:
974:
971:
966:
963:
960:
952:
950:
947:
944:
941:
938:
937:
935:
930:
927:
922:
918:
914:
911:
888:
885:
882:
879:
876:
853:
848:
844:
840:
837:
807:
804:
801:
798:
795:
790:
786:
782:
777:
773:
749:
746:
743:
738:
734:
730:
725:
721:
699:
690:graph coloring
685:
682:
681:
680:
669:
666:
662:
657:
653:
650:
647:
644:
638:
634:
631:
628:
625:
620:
616:
612:
609:
606:
603:
600:
596:
591:
587:
584:
581:
575:
571:
568:
551:
538:
532:
531:
520:
517:
514:
509: if
506:
503:
500:
497:
492:
488:
484:
481:
445:
439:
436:
424:
415:
399:
395:
392:
389:
385:
381:
378:
369:
366:
335:
332:
314:
297:pancake number
271:-pancake graph
261:
242:
241:
235:
234:
229:
224:
220:
219:
190:
186:
185:
182:
176:
175:
169:
163:
162:
159:
153:
152:
145:
139:
138:
126:
122:
119:
116:
112:
108:
105:
96:
93:
80:
74:
73:
62:
59:
49:
43:
42:
37:
33:
31:
23:
22:
13:
10:
9:
6:
4:
3:
2:
4024:
4013:
4010:
4009:
4007:
3992:
3989:
3983:
3980:
3975:
3971:
3966:
3961:
3957:
3953:
3946:
3943:
3938:
3934:
3929:
3924:
3920:
3916:
3909:
3906:
3901:
3897:
3892:
3887:
3883:
3879:
3872:
3869:
3865:
3861:
3857:
3851:
3848:
3843:
3839:
3834:
3829:
3825:
3821:
3814:
3811:
3805:
3800:
3796:
3792:
3788:
3781:
3778:
3774:
3763:on 2012-02-14
3762:
3758:
3752:
3749:
3743:
3738:
3734:
3730:
3729:
3724:
3720:
3716:
3710:
3707:
3702:
3696:
3688:
3686:9780262062824
3682:
3678:
3671:
3668:
3662:
3657:
3653:
3649:
3645:
3638:
3635:
3629:
3624:
3620:
3616:
3612:
3605:
3603:
3599:
3594:
3587:
3580:
3577:
3571:
3566:
3562:
3558:
3554:
3547:
3545:
3541:
3535:
3530:
3523:
3520:
3516:(3): 289–292.
3515:
3511:
3504:
3497:
3495:
3491:
3485:
3480:
3476:
3472:
3468:
3461:
3459:
3457:
3453:
3448:
3444:
3440:
3436:
3431:
3426:
3422:
3418:
3411:
3408:
3403:
3397:
3393:
3389:
3385:
3377:
3374:
3367:
3365:
3361:
3359:
3355:
3351:
3347:
3346:Cayley graphs
3339:
3321:
3314:
3292:
3288:
3285:
3282:
3266:
3252:
3249:
3246:
3241:
3237:
3229:
3226:
3210:
3206:
3202:
3195:
3192:
3191:
3176:
3169:
3155:
3148:
3134:
3131:
3128:
3123:
3119:
3111:
3108:
3092:
3088:
3084:
3077:
3074:
3073:
3058:
3051:
3029:
3026:
3011:
2997:
2994:
2987:
2984:
2968:
2964:
2960:
2953:
2950:
2949:
2934:
2927:
2913:
2910:
2907:
2900:
2886:
2883:
2876:
2873:
2857:
2853:
2849:
2842:
2839:
2838:
2834:
2831:
2828:
2825:
2822:
2819:
2818:
2812:
2809:
2801:
2785:
2782:
2779:
2771:
2768:
2760:
2756:
2749:
2742:
2741:
2740:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2684:
2680:
2676:
2668:
2667:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2616:
2609:
2608:
2604:
2599:
2590:
2576:
2573:
2570:
2550:
2547:
2544:
2541:
2538:
2534:
2530:
2526:
2522:
2519:
2515:
2511:
2508:
2500:
2496:
2491:
2477:
2457:
2454:
2451:
2446:
2442:
2433:
2416:
2412:
2408:
2398:
2396:
2392:
2388:
2384:
2380:
2376:
2368:
2366:
2362:
2360:
2356:
2351:
2330:
2310:
2306:
2301:
2299:
2295:
2290:
2269:
2248:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2173:
2169:
2161:
2160:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2094:
2087:
2086:
2082:
2077:
2068:
2066:
2058:
2056:
2040:
2036:
2033:
2008:
2001:
1998:
1995:
1992:
1989:
1984:
1980:
1976:
1973:
1968:
1964:
1957:
1954:
1943:
1938:
1934:
1929:
1912:
1909:
1906:
1900:
1897:
1894:
1871:
1868:
1865:
1859:
1856:
1848:
1843:
1839:
1834:
1818:
1814:
1811:
1800:
1795:
1791:
1786:
1784:
1768:
1765:
1762:
1759:
1756:
1753:
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321:regular graph
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256:pancake graph
253:
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3761:the original
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2826:Explanation
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252:graph theory
248:mathematical
245:
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3348:, thus are
2499:Manuel Blum
1783:Hamiltonian
534:The γ(
197:Hamiltonian
3767:2008-11-10
3534:2306.11295
3368:References
3358:hypercubes
2776:, if
2595:(sequence
2305:Bill Gates
2073:(sequence
2055:of those.
1742:of length
189:Properties
3960:CiteSeerX
3923:CiteSeerX
3886:CiteSeerX
3833:1111.0434
3735:: 47–57.
3715:Gates, W.
3695:cite book
3430:1201.0452
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