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Studying the Problems Related to Latin Bitrades and 2-Simultaneous Coloring

Sharifzadeh, Maryam | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41682 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoodian, Ebadollah
  7. Abstract:
  8. In the year 1997, at 16th British Combinatorial Conference Cameron introduced a new concept called 2-simultaneous coloring. He used this concept to reformulate a conjecture of Keedwell on the existence of critical partial Latin squares of a given type. After that, other people studied special cases of this conjecture. They also studied the relation between this conjecture and other well-known conjectures in Combinatorics, such as Cycle Double Cover Conjecture (CDC). Finally, in 2004 the general case of this conjecture was proven. In first two sections, in addition to defining some fundamental concepts, we represent the relation between Latin trade and 2-simultaneous coloring. Afterwards, by using the represented materials, we show that Keedwell conjecture and Cameron conjecture are equal. Then, we introduce two concepts in Combinatorics and Graph Theory, which are known as flow and cycle cover. After that, we study the relation between keedwell-Cameron conjecture and another major conjecture in Combinatorics ,which was introduced about 35 years ago, known as Cycle Double Cover Conjecture (CDC). We also study the relation between this conjecture and flows, because of the importance of this topic in algorithmic proof of Keedwell-Cameron conjecture. In the last section, by the use of these materials, we present an algorithmic proof for Keedwell-Cameron conjecture. We also verified our algorithm by the help of a software which is known as “Sage”.
  9. Keywords:
  10. Latin Bitrade ; Cycle Double Cover (CDC) ; Integer Flows ; K-Flows ; 2-Simultaneous Coloring ; Cycle Cover

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