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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 52527 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Daneshgar, Amir
- Abstract:
- Density (alias general connectivity) and sparseness are two fundamental notions in mathematics as well as in graph theory. In this thesis we concentrate on Ne{s}et{r}il's Pentagon problem that asks whether there exists a threshold g{0} such that any cubic graph of girth larger than g{0} is homomorphic to the five cycle. We try to explain why this problem is interesting as a question asking the effect of interplay between two contradictory sparseness and density parameters. To be more explicit, we have dedicated the first two chapters to explain and analysis such situations in general and, in particular, in graph theory by providing and explaining similar statements as deep graph-theoretic conjectures. Also, in the rest of the thesis we have tried to survey different aspects of the pentagon problem, presenting some positive and some negative results, showing the sensitivity of parameters involved, as well as providing different connections of this problem to other important conjectures and facts in the literature. In the last chapter of this thesis we go through some specific constructions of cubic graphs which are based on cylindrical construction technique using Madani's tree-cylinders while we explain the T-graph construction of Daneshgar and Taherkhani in detail
- Keywords:
- Pentagon Problem ; Graph Homomorphism ; Cylindrical Construction
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