Mironenko O. Development of Methods and Algorithms of Solving Complete Graphs T-Factorization Problems

The thesis presents the study of the problem of complete graphs decomposition on trees. Methods and algorithms of solving the problems of complete graphs T-factorization, application of theoretical principles for the construction and enumeration of some groups of non-isomophic decompositions have been analyzed. For each tree T of order 10 possible T-factorization types have been defined, and for the majority of these types realizing T-factorization have been constructed. Non-existence of T-factorization has been proved by the number of regular trees, some stage-and-regular trees and 3-caterpillar factorizations. Basic components of symmetric trees of uneven orders have been constructed that prove the hypothesis of their tree factorization existence. Construction algorithm of all basic multipliers of bicyclical T-factorization for arbitrary values n=4l+2 has been created. A notion of an interfraction line parallel transfer has been introduced. Procedure based on the module excesses and usage of the interfraction line parallel transfer has been elaborated. It allows to calculate all possible T-factorizations for complete graphs Kn with a big number of apexes. The usage of the procedure has been shown on the examples of the complete graphs of order 10 and 14 which have the apex with the highest degree 5 and 7.