Pan-Partition Transitive RealizationsPan-Partition Transitive Realizations

Mike JacobsonUCD

AMS Regional MeetingMiami University

Oxford, Ohio

March 18, 2007

A Primer on Tournaments

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A Primer on Tournaments

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p T(p)

Number of Tournaments

p T(p)

123456789101112131415161718192021222324252627282930

Number of Tournaments

p T(p)

1 12 13 24 45 126 567 4568 688 09 191 53610 9 733 0561112131415161718192021222324252627282930

Number of Tournaments

p T(p)

1 12 13 24 45 126 567 4568 688 09 191 53610 9 733 05611 903 753 24812 154 108 311 16813 48 542 114 686 91214 28 401 423 719 122 30415 31 021 002 160 355 166 84816 63 530 415 842 208 265 100 28817 244 912 778 433 520 759 443 245 82418 1 783 398 846 234 777 975 419 600 287 23219 24 605 641 171 260 376 770 598 003 978 281 47220 645 022 068 557 873 570 931 850 526 424 042 500 09621 32 207 364 031 661 175 384 456 332 260 036 660 040 346 62422 3 070 169 883 150 468 336 193 188 889 176 239 554 269 865 953 28023 559 879 382 429 394 075 397 997 876 821 117 309 031 348 506 639 435 77624 1 956 920 276 575 218 760 843 168 426 608 334 827 851 734 377 775 365 039 898 62425 131 326 696 677 895 002 131 450 257 709 457 767 557 170 027 052 967 027 982 788 816 89626 169 484 335 125 246 268 100 514 597 385 576 342 667 201 246 238 506 672 327 765 919 863 947 26427 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 60828 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 95229 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 51230 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

Number of Tournaments

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

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

2n

2

n!

230

2

30!

Consider n vertices, v1, v2 , v3 , . . . , vn.

Now, consider 30 vertices, v1, v2 , v3 , . . . , v30.

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 thescore sequence of a tournament if and only if

isi1

n

n

2

isi1

k

k

2

for each 1 k < n

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 thescore sequence of a tournament if and only if

isi1

n

n

2

isi1

k

k

2

for each 1 k < n

Bang & Sharp (1982) - gave “the book” proof

Transitive Tournaments

Score Sequence: n-1, n-2, n-3, . . . , 2, 1, 0

“Unique”, acyclic, clear “winner” and/or “loser”,all triples are transitive

Xi

Xi+

1

Xn

X1

Transitive Tournaments

Score Sequence: n-1, n-2, n-3, . . . , 2, 1, 0

“Unique”, acyclic, clear “winner” and/or “loser”,all triples are transitive

Xi

Xi+

1

Xn

X1

Transitive Tournaments

Score Sequence: n-1, n-2, n-3, . . . , 2, 1, 0

“Unique”, acyclic, clear “winner” and/or “loser”,all triples are transitive

Xi+

1

Xi

Xn

X1

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) cyclegives another realization of S.

Ryser (1963) showed that if T1 and T2 are any two realizations ofS, 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.

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;

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 largesttransitive subtournament in any element of T(S).

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 largesttransitive subtournament in any element of T(S).

Define TR(T(S)), abbreviated TR(S), to be the order of the largest largesttransitive subtournament in any element of T(S).

A word about tr(S)… HARD…

Probably…

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).

log2(n) 1 tr(n) 2log2(n) 1

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

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)

Guessing – that finding tr(S) for a general sequence S is going to be hard??

So, what about TR(S)…

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

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!!

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 transitive if 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.

Partition Transitive

Theorem A (Guiduli, Gyárfás, Thomassé &Weidl 1998) For anyscore 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 apartition 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.

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.

Theorem. For any score sequence S, there is a partition transitive realizationinto sets V1 and V2 so that V1 has order TR(S).

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 largesttransitive subtournament in any element of T(S).

Proof

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 isa 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!!

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.

Conjecture. For any score sequence S, and each k, n/2 ≤ k ≤ TR(S) there isa 2-partition transitive realization into parts V1 of order k and V2 of order n-k

Problems/Conjectures

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.