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Non-traditional Round Robin Tournaments

Non-traditional Round Robin Tournaments. Dalibor Froncek University of Minnesota Duluth Mariusz Meszka University of Science and Technology Krak ów. 1–factorization of complete graphs. the complete graph K 2 n : 2 n vertices, every two joined by an edge

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Non-traditional Round Robin Tournaments

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  1. Non-traditional Round Robin Tournaments Dalibor Froncek University of Minnesota Duluth Mariusz Meszka University of Science and Technology Kraków

  2. 1–factorization of complete graphs • the complete graph K2n: 2n vertices, every two joined by an edge • 1–factor: set of n independentedges • 1–factorization: a partition of the edge set of K2n into 2n–1 1–factors

  3. 1–factorization of complete graphs Most familiar 1–factorization of a complete graph K2n: Kirkman, 1846 • geometric construction • labeling construction

  4. 1–factorization of complete graphs Most familiar 1–factorization of a complete graph K2n: Kirkman, 1846 • geometric construction • labeling construction

  5. 1–factorization of complete graphs Most familiar 1–factorization of a complete graph K2n: Kirkman, 1846 • geometric construction • labeling construction

  6. 1–factorization of complete graphs Most familiar 1–factorization of a complete graph K2n: Kirkman, 1846 • geometric construction • labeling construction

  7. 1–factorization of complete graphs Most familiar 1–factorization of a complete graph K2n: Kirkman, 1846 • geometric construction • labeling construction

  8. 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 Round robin tournaments

  9. 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 Round robin tournaments

  10. 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 Round robin tournaments

  11. 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 Round robin tournaments

  12. 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 Round robin tournaments

  13. STEINER Another starter for labeling • Kirkman: 18, 27, 36, 45 • Steiner: 18, 26, 34, 57

  14. Bipartite fact K8 R-B Another factorization: First decompose into two factors, K4,4a 2K4.

  15. Bipartite fact K8 F1 Another factorization: First decompose into two factors, K4,4a 2K4. Then factorize K4,4

  16. Bipartite F1 F2 Another factorization: First decompose into two factors, K4,4a 2K4. Then factorize K4,4

  17. Bipartite F2 F3 Another factorization: First decompose into two factors, K4,4a 2K4. Then factorize K4,4 .

  18. Bipartite F3 F4 Another factorization: First decompose into two factors, K4,4a 2K4. Then factorize K4,4

  19. Bipartite 2K4 Another factorization: First decompose into two factors, K4,4a 2K4. Then factorize K4,4 and finally factorize 2K4.

  20. Bipartite fact K8 R-B Another factorization: First decompose into two factors, K4,4a 2K4. Schedules of this type are useful for two-divisional leagues (like the (in)famous XFL scheduled by J. Dinitz and DF)

  21. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  22. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  23. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  24. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  25. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  26. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  27. “Just run it through a computer!” Number of non-isomorphic 1-factorizationsof the graphKn: n = 4, 6 f = 1 n = 8 f = 6 n = 10 f = 396 n = 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

  28. What is important: • opponent – determined by factorization • in seasonal tournaments (leagues) – home and away games (also determined by factorization)

  29. 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.

  30. Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks.

  31. Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks. Proof: Pigeonhole principle • HAHAHAHA... • HAHAHAHA... • AHAHAHAH...

  32. Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks. Proof: Pigeonhole principle • HAHAHAHA... • AHAHAHAH... • AHAHAHAH...

  33. Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks. Proof: Pigeonhole principle • HAHAHAHA... • AHAHAHAH... • AHAHAHAH...

  34. Lemma: An RRT(2n, 2n–1) has at least 2n–2 breaks. Proof: Pigeonhole principle • HAHAHAHA... • AHAHAHAH... • AHAHAHAH...

  35. 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.

  36. 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

  37. 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

  38. 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

  39. Berger tableswith HAPs team games 1 H 2 H 3 H 4 H 5 A 6 A 7 A 8 A

  40. Berger tableswith HAPs team games 1 HH 2 HA 3 HA 4 HA 5 AA 6 AH 7 AH 8 AH

  41. Berger tableswith HAPs team games 1 HHA 2 HAH 3 HAH 4 HAH 5 AAH 6 AHA 7 AHA 8 AHA

  42. Berger tableswith HAPs team games 1 HHAH 2 HAHH 3 HAHA 4 HAHA 5 AAHA 6 AHAA 7 AHAH 8 AHAH

  43. Berger tableswith HAPs team games 1 HHAHAHA 2 HAHHAHA 3 HAHAHHA 4 HAHAHAH 5 AAHAHAH 6 AHAAHAH 7 AHAHAAH 8 AHAHAHA

  44. Theorem 1: There exists an RRT(2n, 2n–1) with exactly 2n–2 breaks. Proof: Generalize the example for 2n teams.

  45. HOME–AWAY PATTERNS

  46. HOME–AWAY PATTERNS

  47. HOME–AWAY PATTERNS

  48. HOME–AWAY PATTERNS WITH THE SCHEDULE R 1 R 2 R 3 R 4 8–1 5–8 8–2 6–8 7–2 4–6 1–3 5–7 6–3 3–7 7–4 4–1 5–4 2–1 6–5 3–2 R 5 R 6 R 7 8–3 7–8 8–4 2–4 6–1 3–5 1–5 5–2 2–6 7–6 4–3 1–7

  49. Problem: How to schedule an RRT(2n–1, 2n–1)?

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