The Hamiltonian Problems Of Y₃V₃-Free Graphs

Hamitonian Problems is one of the hot areas of the researches of Graph Theory.And this tendency can be seen from the large amount of articles on this field on the key international mathematics magazines and as well as the domestic key mathematics magazines.Determining conditions under which a graph contains a spanning cycle is so called Hamitonian Problems.The Forbidden Subgraphs Problem is one of the most important research fields on Hamiltonain Problems . The claw-free graphs is the most well researched graphs related with Forbidden Subgraphs.Nowadays there are dozens of good results on claw-free graphs.And furthermore,the larger class of graphs related with claw-free graphs such as almost claw-free graphs,quasi claw-free graphs aroused the enthusiasms of many people to focus their researches on this field. Although the new results on the latter two kinds of graphs apears one after another,the research is far from the end.In this paper,we researched the Hamitonian Problems on the Y₃ V₃-free graphs at the first time.The Y₃ V₃-free graphs is a class of graphs larger than the claw-free graphs.The results of this paper extended the Hamitonian Problems research .Firstly we researched the Hamitonian propeties of the Y₃ V₃-free graphs under the conditions that connected and local connected.There are examples show a certain Y₃ V₃-free graphs maybe nonhamiltonian on the conditions metioned above.Furthermore ,there exists Y₃ V₃-free graphs with arbitrarily high connectivity and local connectivity which are nonhamiltonian. The research of the longest cycles will promote the Hamitonian Problem research.Further research on the longest cycles of Y₃ V₃-free graphs got the first important results of this paper:Theorem 1.1.1 If is G a connected and local connected Y₃ V₃-free graph of order...