Whist tournaments – three person property. (English) Zbl 0793.90111
A whist tournament on \(v\) players, denoted by \(Wh(v)\), is a resolvable \((v,4,3)\)-BIBD such that each block \((a,b,c,d)\) represents a game in which the partnership \(a-c\) opposes the partnership \(b-d\), subject to the conditions that each player partners every other player exactly once and opposes every other player exactly twice. \(Wh(v)\) are known to exist for all \(v\) congruent to \(0,1\pmod 4\), \(v\) greater than or equal to 4. A \(Wh(v)\) is said to have the Three Person Property if the intersection of any two games is at most two (2). Such \(Wh(v)\) can be denoted by \(3P- Wh(v)\). In this paper it is proved that every \(Wh(8)\) and every \(Wh(9)\) is a \(3P-Wh(v)\). Specific examples of \(3p-Wh(v)\) are 12,13,15,20,22,24,25,29, and 30. In addition to these examples theorems are proven which lead to the existence of several infinite classes of \(3p-Wh(v)\).
Many of these theorems are established using known results related to PBDs. On the other hand two theorems of N. S. Mendelsohn [in: Recent Progress Combinatorics, Proc. 3rd Waterloo Conference 1968, 123- 132 (1969; Zbl 0192.333)] are also exploited to produce infinite classes of \(3p-Wh(v)\). The paper also establishes explicit formulas that generate two infinite classes of resolvable perfect \((v,4,1)\)-Mendelsohn Designs.
Many of these theorems are established using known results related to PBDs. On the other hand two theorems of N. S. Mendelsohn [in: Recent Progress Combinatorics, Proc. 3rd Waterloo Conference 1968, 123- 132 (1969; Zbl 0192.333)] are also exploited to produce infinite classes of \(3p-Wh(v)\). The paper also establishes explicit formulas that generate two infinite classes of resolvable perfect \((v,4,1)\)-Mendelsohn Designs.
Reviewer: N.J.Finizio (Kingston, U.S.A.)
MSC:
| 91A46 | Combinatorial games |
| 05B05 | Combinatorial aspects of block designs |
| 91A10 | Noncooperative games |
Citations:
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