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Investigating the Conditions of Graphic and Hypergraphic Sequences

Moshfegh, Peyman | 2016

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 48877 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoodian, Ebadollah
  7. Abstract:
  8. Finding necessary and sufficient conditions for a sequence to be graphic or in more general setting existence of factor with predscribed degree sequence of a general graph and also existence of a polynomial algorithm for finding this factor are classical problems of graph theory. These problems were discussed and solved between 1952 to 1972 by prominent mathematicians such as Erdős, Tutte, Edmonds et al. In this thesis we initially discuss most generalized forms of this problems namely Tutte’s f-factor and Lovasz’ (g; f)-factor theorems. Then some simple generalization of the special cases like Erdős-Gallai and Gale-Ryser theorems are investigated. And by using a form of Lovasz’ (g; f)-factor theorem a theorem on orientation of general hypergraphs is stated and proved. Latest theorem generalizes many well-known theorems of graph theory such as necessary and sufficient conditions for a sequence to be losing or score vector of k-hypertournaments, interval tournaments, tournaments etc. Unfortunately for generalizing this classical problems to hypergraphs (uniform or non-uniform) even in simple cases, neither on finding necessary and sufficient conditions nor from computational complexity aspect, no remarkable progress has been achieved yet. But however some partial results have been obtained. First we survey these results on uniform hypergraphs and then on non-uniform hypergraphs. And we present an approximate algorithm for finding a multi-hypergraph with predescribed degree sequence. Proof of optimality of algorithm’s steps is simplified compared with its original articles. And finally we study the set of k-hypergraphic sequences and its convex hull polytope. First we focus on the vertices and edges of this polytope and we show that some obtained results for graphic sequences can not be generalized to k-hypergraphic sequences (k 3). Then we study facets of mentioned polytope and at the latest section by providing two important counter-examples we show that there is no necessary and sufficient conditions similar to Erdős-Gallai and Gale-Ryser theorems for k-hypergraphic sequences or even for degree sequences of some special k-hypergraph’s factors (k 3)
  9. Keywords:
  10. Orientation ; Polytope Method ; Realization ; Factor Analysis ; Graphic Sequence ; Non-Uniform Hypergraphic Sequence ; Uniform Hypergraphic Sequence

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