The Strength of Almost Regular Tournaments C H

The Strength of Almost Regular Tournaments C. H. Hung, Y. L. Wang and J. L. Guo

Outline ․The Story ․Dark Time ․Inspiration ․Composition

The Story

x-Pancircuitous: there is no circuit of length x+1 1 5 2 3 4 7 -pancircuitous

1 1 5 2 3 4 6 2 5 3 4

Almost Regular Tournaments (ART) 1 1 6 5 2 2 3 4 5 3 4

Regular -partite Subtournaments 2 1 3 6 4 5 Professor Chang said: You have to prove its existence.

Is there a circuit of length C(n, 2)-n/2 in an almost regular tournament? 1 6 2 5 3 4 Professor Liu said: Your proof is incorrect.

1 starting 6 2 ending 5 3 4

Dark Time

The High/Low Score Partition of ARTs 2 1 3 6 4 5 1 2 3 4 5 6

The High/Low Score Partition of ARTs 2 1 3 6 4 5 1 2 3 4 5 6

The strength of : the maximum number of matching arcs 1 2 3 4 5 6

The strength of an almost regular tournament is always ?

Inspiration

A counterexample found by Professor Hung. T 5 T 3 T 5

Suppose that there is an ART whose strength is less than.

If there is an ART less than , then with strength.

-tournaments: Properties: S is a regular tournament. H is a regular tournament. is an ART.

-tournament:

-tournament: S is a regular tournament.

-tournament: S is a regular tournament. a contradiction.

-tournament: S and H are regular tournaments.

-tournament: is an ART.

R 1 L 2 R 2

The strength of -tournament R 1 L 2 is R 2 .

L 1 L 2 R 1 R 2