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Critical Sets in Latin Squares

Rahmani Afkham, Mahdi | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41120 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoodian, Ebadollah
  7. Abstract:
  8. In any finite combinatorial configuration, it is possible to identify a subset which uniquely determines the structure of the configuration. Because of their numerous applications in combinatorics, they have a special importance and have been studied for instance in Latin squares, combinatorial designs, and graphs. These sets have been called critical sets in Latin squares, defining sets in combinatorial designs, and forcing sets in different concepts in graphs. In this thesis, we review some papers about critical sets in Latin squares. In Chapter 1, we review the necessary definitions and several theorems and lemmas, for studying critical sets. In Chapter 2, we review the use of Latin bitrades in obtaining several linear and nonlinear lower bounds for the smallest size of critical sets of × Latin squares (scs()). In Chapter 3, we define a notion of "sum" on Latin bitrades, and we review the application of two quite independent combinatorial and algebraic methods in obtaining two algorithms for writing any Latin bitrade as sum of smallest Latin bitrades (intercalates). In Chapter 4, we review this fact that by defining a Map on × Latin squares, we can correlate it with sum of Latin bitrades and intercalates, and then showing that by use of consecutive sums of Latin bitrades with an × Latin square, we can reach to any other × Latin square
  9. Keywords:
  10. Critical set ; Defining Set ; Forcing Sets ; Latin Bitrade ; Latin Square Method ; Intercalate

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