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An Algorithmic Approach to the Intersection Problem in Latin Squares

Mohammadi Nevisi, Maysam | 2009

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 39721 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoudian, Ebadollah
  7. Abstract:
  8. The so-called intersection problem has been considered for many different combinatorial structures, including Latin squares. This intersection problem basically takes a pair of structures, with the same parameters and based on the same underlying set, and determines the possible number of common sub-objects (such as blocks, entries, etc.) which they may have. The intersection problem has also been extended from consideration of pairs of combinatorial structures to sets of three, or even sets of , where may be larger than 3. In this thesis, we study the related problem of determining, for all orders n, the set of integers k for which there exists = 4 Latin squares of order n having precisely k identical cells, and having the four latin squares each containing a different symbol in each of the nnk remaining cells. Fu (1980) solved the problem for = 2, and Adams and etc. (2002) solved the problem for = 3. As there are some constructive methods that generate -way Latin squares recursively based on the smaller ones, finding these small -way Latin squares are of great importance.In this thesis, we have an algorithmic approach to this problem and try to find some 4-way Latin squares of orders less than 15 by computerized programs. We also limit the search space of the problem (i) by proving some conditions of the solutions in each case, and (ii) by showing that in each case, every solution can be manipulated to meet our limiting conditions
  9. Keywords:
  10. Backtracking Method ; Intersection Proplem ; R-Way Latin Square ; Shortcuts

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