Research On Some Problems In Graph Theory

Graph theory is an important branch of mathematics and can be applied widely to many aspects such as computer science and network theory. Hamilton problems are very important on graph theory. At the same time, there are many reasearch on Edge-Pancyclicity of graph. In the present study, a sufficient condition for existence of Hamilton path and results for Edge-Pancyclicity of Varietal Hypercubes are obtained.The main contributions can be summarized as follows:(1) Improvement of criteria for existence of Hamilton path:Let G be a simple connected graph with n(n odd) nodes, if sum of degrees for each pair nodes in G is equal or greater than n-2, then there is a Hamilton path existed.(2) Network of Varietal Hypercubes with n(n≥2) nodes are Edge-Pancyclicy, which contain cycles of length from 4 to 2n.And for n=3k,any edge of Varietal Hypercubes is contained by cycles of length from 4 to 2"Moreover, there is a review for development and application of graph theory before introduction for background and main content of our study. Lastly, basic concepts, symbols and proposition, which are closely related to our study, are also introduced.