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DeFining and Critical Sets of Graphs and Latin Squares

Neshati, Ali | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 47177 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoodian, Ebadollah
  7. Abstract:
  8. Since critical and defining sets have been examined in various fields of combinatorics, including Latin squares and different branches of graph theory, we aim to present a portion of the obtained results con-cerning defining sets in diverse fields of graph theory, and provide a survey on them.In the present dissertation, we examine defining sets of vertex coloring,edge coloring, total coloring and forcing sets of orientation of graphs,perfect matching, hull and geodetic sets. Since complete graphs’ col-oring defining sets and Latin squares’ critical sets are closely related,we shall discuss them in a coloring framework. Furthermore, for dif-ferent graph families, including regular graphs, graphs obtained by Cartesian product of two graphs, Harary graphs and graphs obtained by applying Mycielski operator, we examine defining vertex coloring number. Besides, we comment on the concept of strong and general- ized defining set from vertex coloring defining sets, and elaborate on these two concepts for some families of graphs.At the end, we discuss forcing sets in relation with perfect match-ing, orientation of graphs, and briefly comment on geodetic and hull forcing set and number. we examine perfect matching forcing set and number for diverse families of graphs, such as grid and hypercube graphs, and present hitherto obtained results concerning unilateral and strong orientation forcing sets
  9. Keywords:
  10. Critical set ; Defining Set ; Orientation ; Perfect Matching ; Geodetic Set ; COLORING ; Forcing Sets ; Latin Square Graph ; Hull Set

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