Pan-Partition Transitive Realizations

1. Pan-Partition Transitive Realizations AMS Regional Meeting Miami University Oxford, Ohio March 18, 2007 Mike Jacobson UCD

2. A Primer on Tournaments 1 3 0 4 5 2

3. A Primer on Tournaments 1, 4 3, 2 0, 5 4, 1 5, 0 2, 3

4. Number of Tournaments p T(p)

5. Number of Tournaments p T(p) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

6. Number of Tournaments p T(p) 1 1 2 1 3 2 4 4 5 12 6 56 7 456 8 688 0 9 191 536 10 9 733 056 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

7. Number of Tournaments p T(p) 1 1 2 1 3 2 4 4 5 12 6 56 7 456 8 688 0 9 191 536 10 9 733 056 11 903 753 248 12 154 108 311 168 13 48 542 114 686 912 14 28 401 423 719 122 304 15 31 021 002 160 355 166 848 16 63 530 415 842 208 265 100 288 17 244 912 778 433 520 759 443 245 824 18 1 783 398 846 234 777 975 419 600 287 232 19 24 605 641 171 260 376 770 598 003 978 281 472 20 645 022 068 557 873 570 931 850 526 424 042 500 096 21 32 207 364 031 661 175 384 456 332 260 036 660 040 346 624 22 3 070 169 883 150 468 336 193 188 889 176 239 554 269 865 953 280 23 559 879 382 429 394 075 397 997 876 821 117 309 031 348 506 639 435 776 24 1 956 920 276 575 218 760 843 168 426 608 334 827 851 734 377 775 365 039 898 624 25 131 326 696 677 895 002 131 450 257 709 457 767 557 170 027 052 967 027 982 788 816 896 26 169 484 335 125 246 268 100 514 597 385 576 342 667 201 246 238 506 672 327 765 919 863 947 264 27 421 255 599 848 131 447 082 003 884 098 323 929 861 369 544 621 589 389 269 735 693 986 231 100 612 608 28 2 019 284 625 667 208 265 086 928 694 043 799 677 058 780 746 074 756 618 649 807 453 554 008 410 636 526 845 952 29 18 691 296 182 213 712 407 784 892 577 100 643 237 772 159 079 535 345 610 331 272 616 359 410 643 727 554 822 061 146 512 30 334 493 774 260 141 796 028 606 267 674 709 437 232 608 940 215 918 926 763 659 414 050 175 507 824 571 200 950 884 097 540 096 000

8. Consider n vertices, v1, v2 , v3 , . . . , vn. Now, consider 30 vertices, v1, v2 , v3 , . . . , v30. 334 493 774 260 104 102 715 593 766 508 469 331 712 364 208 913 311 144 737 971 969 121 910 864 452 626 959 027 797 528 575 878 038 T(30) 334 493 774 260 141 796 028 606 267 674 709 437 232 608 940 215 918 926 763 659 414 050 175 507 824 571 200 950 884 097 540 096 000

9. Given a tournament, we can look at the sequence of out-degrees (scores) at each vertex. Call this the score sequence of the tournament. Theorem (Landau 1953) S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn is the score sequence of a tournament if and only if

10. Given a tournament, we can look at the sequence of out-degrees (scores) at each vertex. Call this the score sequence of the tournament. Theorem (Landau 1953) S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn is the score sequence of a tournament if and only if Bang & Sharp (1982) - gave “the book” proof

11. Transitive Tournaments Score Sequence: n-1, n-2, n-3, . . . , 2, 1, 0 Xn “Unique”, acyclic, clear “winner” and/or “loser”, all triples are transitive Xi+1 Xi X1

12. Transitive Tournaments Score Sequence: n-1, n-2, n-3, . . . , 2, 1, 0 Xn “Unique”, acyclic, clear “winner” and/or “loser”, all triples are transitive Xi+1 Xi X1

13. Transitive Tournaments Score Sequence: n-1, n-2, n-3, . . . , 2, 1, 0 Xn “Unique”, acyclic, clear “winner” and/or “loser”, all triples are transitive Xi Xi+1 X1

14. The problem… For a given score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn what can you say about the tournaments that realize the sequence?? SPECIFICALLY -- Given S, what is the order of the largest transitive subtournament over all realizations of S?? Let T(S) be the set of tournaments realizing S; Note, that reversing the arcs on any (directed) cycle gives another realization of S. Ryser (1963) showed that if T1 and T2 are any two realizations of S, then there is a sequence of 3-cycle reversals that will “change” T1 to T2. Brualdi & Li (1982&1984) studied the interchange graph of a sequence S, with vertices labeled with T(S), and two tournaments (vertices) joined by an edge if there is a (single) 3-cycle reversal changing one tournament into the other.

15. The problem… For a given score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn what can you say about the tournaments that realize the sequence?? SPECIFICALLY -- Given S, what is the order of the largest transitive subtournament over all realizations of S?? Let T(S) be the set of tournaments realizing S;

16. The problem… For a given score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn what can you say about the tournaments that realize the sequence?? SPECIFICALLY -- Given S, what is the order of the largest transitive subtournament over all realizations of S?? Let T(S) be the set of tournaments realizing S; Define tr(T(S)), abbreviated tr(S), to be the order of the smallest largest transitive subtournament in any element of T(S).

17. The problem… For a given score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn what can you say about the tournaments that realize the sequence?? SPECIFICALLY -- Given S, what is the order of the largest transitive subtournament over all realizations of S?? Let T(S) be the set of tournaments realizing S; Define tr(T(S)), abbreviated tr(S), to be the order of the smallest largest transitive subtournament in any element of T(S). Define TR(T(S)), abbreviated TR(S), to be the order of the largest largest transitive subtournament in any element of T(S).

18. A word about tr(S)… Probably… HARD… Let T(n) be the set of tournaments on n vertices and let tr(n) be the order of the smallest largest transitive subtournament in any element of T(n). In other words, tr(n) is the smallest integer so that every tournament T of order n contains a transitive subtournament of order tr(n) Theorem (Erdös & Moser 1964) Reid and Parker (1970) showed that the the lower bound isn’t tight Sanchez-Flores (1998) showed that tr(n) ≥ log2(n) + 1.2451 Guessing – that finding tr(S) for a general sequence S is going to be hard??

19. Let T(n) be the set of tournaments on n vertices and let TR(n) be the order of the largest largest transitive subtournament in any element of T(n). TR(n) = n So, what about TR(S)… S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn Find the largest k so that S majorizes k-1, k-2, k-3, . . . , 2, 1, 0 sn ≥ k-1, sn-1 ≥ k-2, sn-2 ≥ k-3, . . . , sn-k-3 ≥ 2, sn-k-2 ≥ 1, sn-k-1 ≥ 0 Unfortunately (??) that is arbitrarily far away from being correct!!

20. Almost there… For a score sequence S, a partition transitive realization, is a tournament T realizing S, so that the vertices of T can be partitioned into sets V1 and V2 so that the subtournaments on V1 and V2 are transitive. For a score sequence S, S is pan -partition transitiveif for each k, n/2 ≤ k ≤ TR(S) there is a tournament T realizing S, so that the vertices of T can be partitioned into sets V1 of order k and V2 of order n-k so that the subtournaments on V1 and V2 are transitive.

21. Partition Transitive Theorem A (Guiduli, Gyárfás, Thomassé &Weidl 1998) For any score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn there is a partition transitive realization with “equal” parts. In fact, they showed, for S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn there is a partition transitive realization with transitive subtournaments on the vertices with the even indexed scores and one on the vertices with the odd indexed scores. Aho and Hanson (1998) gave an independent proof of Theorem A.

22. Partition Transitive Realizations Brualdi and Shen gave an additional proof in 2001 and posed the more general question; for score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn and n/2≥ n1 ≥n2 ≥ n3 ≥ . . . ≥ nk so that n = n1+n2+n3+ . . . +nk then there is a k-partition transitive realization with parts having order ni. Accosta, Bassa, Chaikin, Reihl, Tingstad, Zhao and Kleitman (2003) proved this conjecture.

23. We ask -- why stop at n/2 ?? Joint work with Art Busch, Guantao Chen and Jian Shen Recall, for score sequence S, TR(S) is the order of the largest largest transitive subtournament in any element of T(S). Theorem. For any score sequence S, there is a partition transitive realization into sets V1 and V2 so that V1 has order TR(S). Proof

24. What about pan-partition transitive realizations?? Fix a score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn Let p = | { i | si > i-1 } |, let q = | { i | si < i-1 } |, let z = | { i | si = i-1 } | and let t = max { p+z, q+z } Theorem.For any score sequence S, and each k, n/2 ≤ k ≤ t there is a 2-partition transitive realization into parts V1 of order k and V2 of order n-k Proof…get the paper Which isn’t written – yet!!

25. Problems/Conjectures Conjecture.For any score sequence S, and each k, n/2 ≤ k ≤ TR(S) there is a 2-partition transitive realization into parts V1 of order k and V2 of order n-k Brualdi and Shen - like Conjecture for score sequence S = s1 ≤ s2 ≤ s3 ≤ . . . ≤ sn and TR(S)≥ n1 ≥n2 ≥ n3 ≥ . . . ≥ nk so that n = n1+n2+n3+ . . . +nk there is a k-partition transitive realization with parts having order ni.

26. Visitors needed – a plea… UCD may have a position or two for “post-doc” like visitors, for a year, maybe two, 2 course teaching load (per semester), living wage… So, if you are, have – or know of a student that is still looking for a position, and may be willing to visit for a year, preferably in Graph Theory, but the department will certainly look at combinatorialists (finite geometers), Optimization, OR, P&S, and/or computational math/science, have them contact me at msj@cudenver.edu.