Non-traditional Round Robin Tournaments
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Transcript of Non-traditional Round Robin Tournaments
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Non-traditional Round Robin Tournaments
Dalibor FroncekUniversity of Minnesota Duluth
Mariusz MeszkaUniversity of Science and Technology Kraków
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1–factorization of complete graphs
•the complete graph K2n:2n vertices, every two joined by an edge
•1–factor: set of n independent edges•1–factorization: a partition of the edge set of K2n into 2n–1 1–factors
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1–factorization of complete graphs
Most familiar 1–factorization of a complete graph K2n:
Kirkman, 1846
•geometric construction•labeling construction
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1–factorization of complete graphs
Most familiar 1–factorization of a complete graph K2n:
Kirkman, 1846
•geometric construction•labeling construction
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1–factorization of complete graphs
Most familiar 1–factorization of a complete graph K2n:
Kirkman, 1846
•geometric construction•labeling construction
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4 5
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![Page 7: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/7.jpg)
1–factorization of complete graphs
Most familiar 1–factorization of a complete graph K2n:
Kirkman, 1846
•geometric construction•labeling construction
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4 5
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![Page 8: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/8.jpg)
1–factorization of complete graphs
Most familiar 1–factorization of a complete graph K2n:
Kirkman, 1846
•geometric construction•labeling construction
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Round robin tournaments
Round robin tournament• 2n teams• every two teams play
exactly one game• tournament consists of
2n–1 rounds • each plays exactly one
game in each round
Complete graph• 2n vertices• every two vertices
joined by an edge• K2n is factorized into
2n–1 factors• factors are regular of
degree 1
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Round robin tournaments
Round robin tournament• 2n teams• every two teams play
exactly one game• tournament consists of
2n–1 rounds • each plays exactly one
game in each round
Complete graph• 2n vertices• every two vertices
joined by an edge• K2n is factorized into
2n–1 factors• factors are regular of
degree 1
![Page 11: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/11.jpg)
Round robin tournaments
Round robin tournament• 2n teams• every two teams play
exactly one game• tournament consists of
2n–1 rounds • each plays exactly one
game in each round
Complete graph• 2n vertices• every two vertices
joined by an edge• K2n is factorized into
2n–1 factors• factors are regular of
degree 1
![Page 12: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/12.jpg)
Round robin tournaments
Round robin tournament• 2n teams• every two teams play
exactly one game• tournament consists of
2n–1 rounds • each plays exactly one
game in each round
Complete graph• 2n vertices• every two vertices
joined by an edge• K2n is factorized into
2n–1 factors• factors are regular of
degree 1
![Page 13: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/13.jpg)
Round robin tournaments
Round robin tournament• 2n teams• every two teams play
exactly one game• tournament consists of
2n–1 rounds • each plays exactly one
game in each round
Complete graph• 2n vertices• every two vertices
joined by an edge• K2n is factorized into
2n–1 factors• factors are regular of
degree 1
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STEINERAnother starter for labeling
•Kirkman:18, 27, 36, 45•Steiner:18, 26, 34, 57
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Bipartite fact K8 R-B
Another factorization:First decompose into two factors, K4,4 a 2K4.
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Bipartite fact K8 F1
Another factorization:First decompose into two factors, K4,4 a 2K4.
Then factorize K4,4
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Bipartite F1 F2
Another factorization:First decompose into two factors, K4,4 a 2K4.
Then factorize K4,4
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Bipartite F2 F3
Another factorization:First decompose into two factors, K4,4 a 2K4.
Then factorize K4,4
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Bipartite F3 F4
Another factorization:First decompose into two factors, K4,4 a 2K4.
Then factorize K4,4
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Bipartite 2K4
Another factorization:First decompose into two factors, K4,4 a 2K4.
Then factorize K4,4
and finally factorize 2K4.1
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Bipartite fact K8 R-B
Another factorization:First decompose into two factors, K4,4 a 2K4.
Schedules of this type are useful for two-divisional leagues(like the (in)famous XFLscheduled by J. Dinitz and DF)
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“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
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“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
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“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
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“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
![Page 26: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/26.jpg)
“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
![Page 27: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/27.jpg)
“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
![Page 28: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/28.jpg)
“Just run it through a computer!”
Number of non-isomorphic 1-factorizations of the graph Kn:
n = 4, 6 f = 1n = 8 f = 6n = 10 f = 396n = 12 f = 526 915 620 (Dinitz, Garnick, McKay, 1994)
Number of different schedules for 12 teams: 1 346 098 266 906 624 000
Estimated number of schedules for 16 teams: 1058
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What is important:
• opponent – determined by factorization• in seasonal tournaments (leagues) – home
and away games (also determined by factorization)
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Ideal home-away pattern (HAP):
Ideally either•HAHAHAHA... or•AHAHAHAH...
Unfortunately, there can be at most two teams with one of these ideal HAPs.
A subsequence AA or HH is called a break in the HAP.
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Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.
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Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.
Proof:Pigeonhole principle•HAHAHAHA...•HAHAHAHA...•AHAHAHAH...
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Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.
Proof:Pigeonhole principle
•HAHAHAHA...•AHAHAHAH...•AHAHAHAH...
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Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.
Proof:Pigeonhole principle
•HAHAHAHA...•AHAHAHAH...•AHAHAHAH...
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Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.
Proof:Pigeonhole principle
•HAHAHAHA...•AHAHAHAH...•AHAHAHAH...
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Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.
We will now show that schedules with this number of breaks really exist.
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Kirkman factorization of K8 – Berger tables
• Round 1 – factor F1
• Round 2 – factor F5
• Round 3 – factor F2
• Round 4 – factor F6
• Round 5 – factor F3
• Round 6 – factor F7
• Round 7 – factor F4
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Kirkman factorization of K8 – Berger tables
• Round 1 – factor F1
• Round 2 – factor F5
• Round 3 – factor F2
• Round 4 – factor F6
• Round 5 – factor F3
• Round 6 – factor F7
• Round 7 – factor F4
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Kirkman factorization of K8 – Berger tables
• Round 1 – factor F1
• Round 2 – factor F5
• Round 3 – factor F2
• Round 4 – factor F6
• Round 5 – factor F3
• Round 6 – factor F7
• Round 7 – factor F4
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Berger tables with HAPs
team games1 H2 H3 H4 H5 A6 A7 A8 A
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Berger tables with HAPs
team games1 HH2 HA3 HA4 HA5 AA6 AH7 AH8 AH
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Berger tables with HAPs
team games1 HHA2 HAH3 HAH4 HAH5 AAH6 AHA7 AHA8 AHA
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Berger tables with HAPs
team games1 HHAH2 HAHH3 HAHA4 HAHA5 AAHA6 AHAA7 AHAH8 AHAH
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Berger tables with HAPs
team games1 HHAHAHA2 HAHHAHA3 HAHAHHA4 HAHAHAH5 AAHAHAH6 AHAAHAH7 AHAHAAH8 AHAHAHA
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Theorem 1: There exists an RRT(2n, 2n–1) with exactly 2n–2 breaks.
Proof: Generalize the example for 2n teams.
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HOME–AWAY PATTERNS
R 1 R 2 R 3 R 4 R 5 R 6 R 71 H H A H A H A2 H A H H A H A3 H A H A H H A4 H A H A H A H5 A A H A H A H6 A H A A H A H7 A H A H A A H8 A H A H A H A
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HOME–AWAY PATTERNS
R 1 R 2 R 3 R 4 R 5 R 6 R 71 H H A H A H A2 H A H H A H A3 H A H A H H A4 H A H A H A H5 A A H A H A H6 A H A A H A H7 A H A H A A H8 A H A H A H A
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HOME–AWAY PATTERNS
R 1 R 2 R 3 R 4 R 5 R 6 R 71 H H A H A H A2 H A H H A H A3 H A H A H H A4 H A H A H A H5 A A H A H A H6 A H A A H A H7 A H A H A A H8 A H A H A H A
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HOME–AWAY PATTERNS WITH THE SCHEDULE
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H A H A
2 H A H H A H A
3 H A H A H H A
4 H A H A H A H
5 A A H A H A H
6 A H A A H A H
7 A H A H A A H
8 A H A H A H A
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Problem: How to schedule an RRT(2n–1, 2n–1)?
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Problem: How to schedule an RRT(2n–1, 2n–1)?
Equivalent problem: How to catch 2n–1 lions?
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Problem: How to schedule an RRT(2n–1, 2n–1)?
Equivalent problem: How to catch 2n–1 lions?
Solution: Catch 2n of them and release one.
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Problem: How to schedule an RRT(2n–1, 2n–1)?
Solution: Schedule a RRT(2n, 2n–1). Then select one team to be the dummy team. That means, whoever is scheduled to play the dummy team in a round i has a bye in that round.
We only need to be careful to select the right dummy team.
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Select the Dummy Team
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H A H A
2 H A H H A H A
3 H A H A H H A
4 H A H A H A H
5 A A H A H A H
6 A H A A H A H
7 A H A H A A H
8 A H A H A H A
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Select the Dummy Team
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H A H A
2 H A H H A H A
3 H A H A H H A
4 H A H A H A H
5 A A H A H A H
6 A H A A H A H
7 A H A H A A H
8 A H A H A H A
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Dummy Team = 5
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H A H A
2 H A H H A H A
3 H A H A H H A
4 H A H A H A H
5 A A H A H A H
6 A H A A H A H
7 A H A H A A H
8 A H A H A H A
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Dummy Team = 5
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H H A
2 H A H H A A
3 H A H A H H
4 A H A H A H
5
6 A H A H A H
7 A H A A A H
8 A A H A H A
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Dummy Team = 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2
3 H A H H H A
4 H A H A A H
5 A A H A H H
6 A H A A H A
7 H A H A A H
8 A H H A H A
![Page 59: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/59.jpg)
Dummy Team = 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2
3 H A H H H A
4 H A H A A H
5 A A H A H H
6 A H A A H A
7 H A H A A H
8 A H H A H A
![Page 60: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/60.jpg)
Dummy Team = 8
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H A H A
2 H A H H A H A
3 H A H A H H A
4 H A H A H A H
5 A A H A H A H
6 A H A A H A H
7 A H A H A A H
8 A H A H A H A
![Page 61: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/61.jpg)
Dummy Team = 8
R 1 R 2 R 3 R 47–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 72–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
A schedule with no breaks!
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 62: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/62.jpg)
Given the HAP, find a schedule
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 63: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/63.jpg)
Look at the game between 1 and 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 64: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/64.jpg)
Look at the game between 1 and 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 65: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/65.jpg)
Look at the game between 1 and 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 66: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/66.jpg)
Look at the game between 1 and 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 67: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/67.jpg)
Look at the game between 1 and 2
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 68: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/68.jpg)
Look at the game between 2 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 69: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/69.jpg)
Look at the game between 2 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 70: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/70.jpg)
Look at the game between 2 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 71: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/71.jpg)
Look at the game between 2 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 72: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/72.jpg)
Look at the game between 2 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 73: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/73.jpg)
Look at the game between 3 and 4
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 74: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/74.jpg)
And so on…
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 75: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/75.jpg)
One more step – teams 1 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 76: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/76.jpg)
One more step – teams 1 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 77: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/77.jpg)
One more step – teams 1 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 78: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/78.jpg)
One more step – teams 1 and 3
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 79: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/79.jpg)
One more step – teams 2 and 4
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 80: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/80.jpg)
And so on…
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 81: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/81.jpg)
One more time – teams 1 and 4
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 82: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/82.jpg)
One more time – teams 2 and 5
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 83: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/83.jpg)
…and we are done!
R 1 R 2 R 3 R 48–1 5–8 8–2 6–87–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 78–3 7–8 8–42–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 84: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/84.jpg)
Theorem 2: There exists an RRT(2n –1, 2n–1) with no breaks. Moreover, this schedule is unique.
Proof: Use the ideas from the example.
![Page 85: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/85.jpg)
RRT(2n,2n) – a schedule for 2n teams in 2n weeks.Every team has exactly one bye.
![Page 86: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/86.jpg)
RRT(2n,2n) – a schedule for 2n teams in 2n weeks.Every team has exactly one bye.
Who needs such a schedule anyway??
![Page 87: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/87.jpg)
University of Vermont — Men’s Basketball 2002–2003
JAN Thu 2 at Maine* Sun 5 STONY BROOK* Wed 8 NORTHEASTERN* Sat 11 at Boston University* Mon 13 at CornellWed 15 at Albany*Wed 22 MAINE* Sat 25 HARTFORD* Wed 29 at New Hampshire*
FEB Sun 2 at Binghamton* Tue 4 MIDDLEBURY Sat 8 at Stony Brook* Wed 12 BINGHAMTON* Sat 15 at Northeastern*Wed 19 NEW HAMPSHIRE* Sat 22 BOSTON UNIVERSITY* Wed 26 at Hartford* MAR Sun 2 ALBANY*
![Page 88: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/88.jpg)
University of Vermont — Men’s Basketball 2002–2003
JAN Thu 2 at Maine* Sun 5 STONY BROOK* Wed 8 NORTHEASTERN* Sat 11 at Boston University* Mon 13 at CornellWed 15 at Albany*Wed 22 MAINE* Sat 25 HARTFORD* Wed 29 at New Hampshire*
FEB Sun 2 at Binghamton* Tue 4 MIDDLEBURY Sat 8 at Stony Brook* Wed 12 BINGHAMTON* Sat 15 at Northeastern*Wed 19 NEW HAMPSHIRE* Sat 22 BOSTON UNIVERSITY* Wed 26 at Hartford* MAR Sun 2 ALBANY*
![Page 89: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/89.jpg)
Definition: An extended round robin tournament RRT*(n,k) is a tournament RRT(n,k) (with n k) where every bye is replaced by an interdivisional game.
![Page 90: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/90.jpg)
Definition: An extended round robin tournament RRT*(n,k) is a tournament RRT(n,k) (with n k) where every bye is replaced by an interdivisional game.
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H A H A H A
2 H A H A H A
3 H A H A H A
4 H A H A H A
5 A H A H A H
6 A H A H A H
7 A H A H A H
![Page 91: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/91.jpg)
Definition: An extended round robin tournament RRT*(n,k) is a tournament RRT(n,k) (with n k) where every bye is replaced by an interdivisional game.
R 1 R 2 R 3 R 4 R 5 R 6 R 7
1 H H A H A H A
2 H A H H A H A
3 H A H A H H A
4 H A H A H A A
5 A A H A H A H
6 A H A A H A H
7 A H A H A A H
![Page 92: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/92.jpg)
Question: Can we find an RRT*(n,n) with the perfect HAP?
![Page 93: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/93.jpg)
Question: Can we find an RRT*(n,n) with the perfect HAP?
Look at the teams starting HOME.
1 2 3 4 5 6 …
1 H A H A H A …
2 H A H A H A …
3 H A H A H A …
![Page 94: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/94.jpg)
Question: Can we find an RRT*(n,n) with the perfect HAP?
Look at the teams starting HOME. They will never meet, no matter when they play their interdivisional games.
1 2 3 4 5 6 …
1 H A H A H A …
2 H A H A H A …
3 H A H A H A …
![Page 95: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/95.jpg)
Question: Can we find an RRT*(n,n) with the perfect HAP?
Look at the teams starting HOME. They will never meet, no matter when they play their interdivisional games.
1 2 3 4 5 6 …
1 H A H A H A …
2 H A H A H A …
3 H A H A H A …
![Page 96: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/96.jpg)
RRT(2n,2n) – a schedule for 2n teams in 2n weeks.Every team has exactly one bye.
We want to prove the following
![Page 97: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/97.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.
![Page 98: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/98.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.Bye teams come in pairs, so suppose there are 4 of them.
… A H A B H A H …
… A H A B H A H …
… H A H B A H A …
… H A H B A H A …
![Page 99: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/99.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.Bye teams come in pairs, so suppose there are 4 of them.
… H A H B A H A …
… H A H B A H A …
![Page 100: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/100.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.Bye teams come in pairs, so suppose there are 4 of them.
… H A H B A H A …
… H A H B A H A …
![Page 101: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/101.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.Bye teams come in pairs, so suppose there are 4 of them.
… H A H B A H A …
… H A H B A H A …
![Page 102: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/102.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 2. There are at most two teams with a bye in any two consecutive rounds..
… A H A B H A H …
… H A H B A H A …
… H A H A B H A …
A H A H B A H
![Page 103: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/103.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 2. There are at most two teams with a bye in any two consecutive rounds..
… A H A B H A H …
… H A H B A H A …
… H A H A B H A …
A H A H B A H
![Page 104: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/104.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 2. There are at most two teams with a bye in any two consecutive rounds..
… A H A B H A H …
… H A H B A H A …
… H A H A B H A …
A H A H B A H
![Page 105: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/105.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.
Claim 2. There are at most two teams with a bye in any two consecutive rounds..
So we are done, and have byes in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
![Page 106: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/106.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.
Claim 2. There are at most two teams with a bye in any two consecutive rounds..
So we are done, and have byes in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
WRONG!!!
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
![Page 107: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/107.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.
Claim 2. There are at most two teams with a bye in any two consecutive rounds..
So we are done, and have byes in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
WRONG!!! What about rounds 1, 3, … , 2n–2, 2n?
![Page 108: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/108.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 3. If there are teams with a bye in Round 1 then there are no teamswith a bye in Round 2n and vice versa.
B A H A … H A H A
B H A H … A H A H
A H A H … A H A B
H A H A … H A H B
![Page 109: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/109.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 3. If there are teams with a bye in Round 1 then there are no teamsWith a bye in Round 2n and vice versa.
B A H A … H A H A
B H A H … A H A H
A H A H … A H A B
H A H A … H A H B
![Page 110: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/110.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 3. If there are teams with a bye in Round 1 then there are no teamsWith a bye in Round 2n and vice versa.
B A H A … H A H A
B H A H … A H A H
A H A H … A H A B
H A H A … H A H B
![Page 111: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/111.jpg)
Theorem 3: In RRT(2n,2n) without breaks there are exactly two teams with a bye in weeks 1, 3,…, 2n–1 or in weeks 2, 4,…, 2n.
Proof:Claim 1. There are at most two teams with a bye in a round. Moreover, these two teams have complementary HAPs.
Claim 2. There are at most two teams with a bye in any two consecutive rounds..
Claim 3. If there are teams with a bye in Round 1 then there are no teamswith a bye in Round 2n and vice versa.
Now we are really done.
![Page 112: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/112.jpg)
Let us find a schedule for RRT(2n,2n)
![Page 113: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/113.jpg)
Let us find a schedule for RRT(2n,2n)
• But how?
![Page 114: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/114.jpg)
Let us find a schedule for RRT(2n,2n)
• But how?• The schedule for RRT(2n–1,2n–1) was uniquely
determined by its HAP.
![Page 115: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/115.jpg)
Let us find a schedule for RRT(2n,2n)
• But how?• The schedule for RRT(2n–1,2n–1) was uniquely
determined by its HAP.
• So let us try what the HAP yields.
![Page 116: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/116.jpg)
By our Theorem 3, a schedule for 12 teams looks like this.
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 117: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/117.jpg)
Look at 1 and 2R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 118: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/118.jpg)
Look at 1 and 2
Both home: cannot play
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 119: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/119.jpg)
Look at 1 and 2
Both home: cannot play
Both away:cannot play
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 120: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/120.jpg)
Look at 1 and 2
Both home: cannot play
Both away:cannot play
So there is just oneround left.
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 121: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/121.jpg)
This way it works for all pairsk, k+1
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 122: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/122.jpg)
Now teams 1 and 3R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 123: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/123.jpg)
Now teams 1 and 3
Cannot play each other when anothergame has already been scheduled
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 124: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/124.jpg)
Now teams 1 and 3
Cannot play each other when anothergame has already been scheduled
Cannot play whenboth have a homegame
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 125: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/125.jpg)
Now teams 1 and 3
Cannot play each other when anothergame has already been scheduled
Cannot play whenboth have a homegame
or both have anaway game
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 126: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/126.jpg)
Now teams 1 and 3
Cannot play each other when anothergame has already been scheduled
Cannot play whenboth have a homegame
or both have anaway game.
So again just one choice.
R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 127: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/127.jpg)
Similarly…R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 128: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/128.jpg)
And again…R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R10 R11 R12
1 H A H A H A H A H A H2 H A H A H A H A H A H3 H A H A H A H A H A H4 H A H A H A H A H A H5 H A H A H A H A H A H6 H A H A H A H A H A H7 A H A H A H A H A H A8 A H A H A H A H A H A9 A H A H A H A H A H A10 A H A H A H A H A H A11 A H A H A H A H A H A12 A H A H A H A H A H A
![Page 129: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/129.jpg)
Theorem 3: There exists a RRT(2n, 2n) with no breaks. Moreover, this schedule is unique.
Proof: Use the ideas from the example.
![Page 130: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/130.jpg)
Another look at our RRT(7, 7)
R 1 R 2 R 3 R 47–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 72–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
aij = k the game between i and j is played in Round k
1 2 3 4 5 6 7
1
2
3
4
5
6
7
![Page 131: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/131.jpg)
Another look at our RRT(7, 7)
R 1 R 2 R 3 R 47–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 72–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
aij = k the game between i and j is played in Round k
1 2 3 4 5 6 7
1 2 3 4 5 6 7
2 2 4 5 6 7 1
3 3 4 6 7 1 2
4 4 5 6 1 2 3
5 5 6 7 1 3 4
6 6 7 1 2 3 5
7 7 1 2 3 4 5
![Page 132: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/132.jpg)
Another look at our RRT(7, 7)
R 1 R 2 R 3 R 47–2 4–6 1–3 5–76–3 3–7 7–4 4–15–4 2–1 6–5 3–2
R 5 R 6 R 72–4 6–1 3–51–5 5–2 2–67–6 4–3 1–7
aii = k
team i has a bye in Round k
1 2 3 4 5 6 7
1 1 2 3 4 5 6 7
2 2 3 4 5 6 7 1
3 3 4 5 6 7 1 2
4 4 5 6 7 1 2 3
5 5 6 7 1 2 3 4
6 6 7 1 2 3 4 5
7 7 1 2 3 4 5 6
![Page 133: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/133.jpg)
Another look at RRT(8, 8)
R 1 R 2 R 3 R 48–2 5–6 1–3 6–77–3 4–7 8–4 5–86–4 3–8 7–5 4–1
2–1 3–2
R 5 R 6 R 7 R 82–4 7–8 3–5 8–11–5 6–1 2–6 7–28–6 5–2 1–7 6–3
4–3 5–4
1 2 3 4 5 6 7 8
1 1 2 3 4 5 6 7 8
2 2 3 4 5 6 7 8 1
3 3 4 5 6 7 8 1 2
4 4 5 6 7 8 1 2 3
5 5 6 7 8 1 2 3 4
6 6 7 8 1 2 3 4 5
7 7 8 1 2 3 4 5 6
8 8 1 2 3 4 5 6 7
![Page 134: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/134.jpg)
A general RRT(n, n)
1 2 3 … n–1 n
1 1 2 3 n–1 n
2 2 3 4 n 1
3 3 4 5 1 2
… …
n–1 n–1 n 1 … n–4 n–3 n–2
n n 1 2 … n–3 n–2 n–1
![Page 135: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/135.jpg)
A general RRT(n, n)
Substitute i i–11 2 3 … n–1 n
1 1 2 3 n–1 n
2 2 3 4 n 1
3 3 4 5 1 2
… …
n–1 n–1 n 1 … n–4 n–3 n–2
n n 1 2 … n–3 n–2 n–1
![Page 136: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/136.jpg)
A general RRT(n, n)
Substitute i i–1
to get the group Zn !!
0 1 2 … … n–3 n–2 n–1
0 0 1 2 n–3 n–2 n–1
1 1 2 3 n–2 n–1 0
2 2 3 4 n–1 0 1
… …
n–2 n–2 n–1 0 … … n–5 n–4 n–3
n–1 n–1 0 1 … … n–4 n–3 n–2
![Page 137: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/137.jpg)
A general RRT(n, n)
Substitute i i–1
to get the group Zn !!
In other words, we getthe table of addition mod n
0 1 2 … … n–3 n–2 n–1
0 0 1 2 n–3 n–2 n–1
1 1 2 3 n–2 n–1 0
2 2 3 4 n–1 0 1
… …
n–2 n–2 n–1 0 … … n–5 n–4 n–3
n–1 n–1 0 1 … … n–4 n–3 n–2
![Page 138: Non-traditional Round Robin Tournaments](https://reader035.fdocuments.in/reader035/viewer/2022062323/56815cb0550346895dcaaec0/html5/thumbnails/138.jpg)
!!!!!!!!!!!!!! THE END !!!!!!!!!!!!