Large Sets Of Hamilton Cycle And Path Decompositions

The large set problem in the combinatorial design theory has a long history and a series ofapplications in experimental design, coding theory, etc.The progress of the related research hadbeen quite slow during a long period of time due to its sophistication. Being benefited and motivatedby some new methods and techniques, the research in the large set problem has taken on a promisingposture in recent twenty years.About large sets of (directed) cycle systems and path decompositions, much work has beendone. Jiaxi Lu [50, 51] and L. Teirlinck [60] gave the existence spectrum for large sets of 3-cycle systems of K_v, that is, large sets of Steiner triple systems. D. Bryant [8] proved that thereexists a large set of Hamilton cycle decomposition of K_2t+1(K_2t-F), and there exists a largeset of Hamilton path decomposition of K_2t(K_2t+1-f).Q.Kang and Y. Zhang [44] obtainedthe existence spectrum for large sets of 3-path decompositions ofλK_v, and obtained some otherresults about large sets of k-path decompositions ofλK_v in [67]. Q. Kang, J. Lei and Y. Chang [40]gave the existence spectrum for large sets of directed 3-cycle systems ofλK_v^*, that is, large sets ofMendelsohn triple systems. Y. Zhang and Q. Kang [68] obtained the existence spectrum for largesets of directed 3-path decompositions ofλK_v^*. Q. Kang [31] completed the existence spectrum forlarge sets of almost directed Hamilton cycle decompositions of K_v^*, and gave some results aboutlarge sets of directed Hamilton cycle and path decompositions of K_v^*.In this dissertation, we discuss mainly the existence problem of large sets of (almost) Hamiltoncycle and path decompositions (including the directed cases).In Chapter 1, we introduce some terminologies and basic concepts, give some known resultsabout cycle systems and path decompositions, and their large sets. We also list the main conclusionscontained in this dissertation.In Chapter 2, we discuss the existence of large sets of Hamilton cycle and path decompositionsofλK_v and obtain their existence spectrums using complete automorphism groups. For large setsof directed Hamilton cycle and path decompositions ofλK_v^*, we give a partial solution using certainspecial Tuscan squares, and also provide some approach to complete the spectrums.In Chapter 3, we research the existence of large sets of almost Hamilton cycle decompositionsofλK_v. We show that the completion of the existence spectrum only depends on one case:λ=2 and v≥4. Furthermore, using symmetric (and alternating) groups, we obtain the existencespectrum except the possible case: v≡3 (mod 4) and v≥15.In Chapters 4, 5, using coset representatives, we obtain the existence spectrums for large setsof Hamilton cycle and path decompositions ofλK_m,n, and large sets of directed Hamilton cycle and path decompositions ofλK_m,n^*. In addition, we obtain the existence spectrum for large sets ofalmost Hamilton cycle and path decompositions ofλK_m,n except one possible infinite family.In Chapter 6, we first give a method to construct large sets of resolvable Mendelsohn triplesystems of order q + 2, where q =6t + 1 is a prime power. Then, by using coloring technique,large sets of resolvable directed triple systems with the same orders are obtained too. Furthermore,using tripling construction, product construction for LRMTSs and LRDTSs, and new results forLR-designs, we obtain more infinite families. Finally, we provide new product construction forLRMTSs and LRDTSs.In Chapter 7, some other problems about designs are researched.