The Role of Trades in 4-Cycle Systems

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Malek, Hamid | 2018

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51511 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Mahmoodian, Abadollah
Abstract:
a graph into specific subgraphs. Specially, decomposing the edges of a complete graph into cycles with determined length draws the main attention. Most of the former works in this area were studied the Alspach conjecture. In another words, Alspach conjecture states that ” Is the obvious necessary conditions for decomposing a graph into cycles (equality of the edges count with sum of cycle lengths) with specified length but not necessarily identical are also sufficient.” Different cases of Alspach conjecture have been studied and proved completely.The decomposition of a complete graph into cycles with the same length k is called k-cycle system. Steiner triple system is a 3-cycle system. Furthermore, many articles have been studied 4-cycle systems.The pair of sets like T and S of cycles with length k on the vertex set V is called a trade of volume jTj and foundation jV j if no cycle in T repeated in S and the edges in cycles of both sets are equal. In this thesis, the 4–cycle systems and the role of trades in constructing 4-cycle systems with constant order and the relationship between them are being studied
Keywords:
4-Cycle Systems ; Trade ; Graph Decomposition ; Alspach Conjecture