The Contractible Edges Of The 6 Connected Graphs

As early as 200 years ago, humans began to get involved in the research field of graph theory. In 1736, with graph, Euler solved Konigsberg Bridge Problem Seven and published his first paper on graph theory. Since 1930s, graph theory became active and extraordinary, highlighting its differences in the scientific community. There are many well-known graph theory problems, such as Hamiltonian problems, four-color problem, the Chinese postman problem, etc. It is also used in biology, chemistry, information and computer science disciplines, which has shown great superiority. Meanwhile, graph theory has been used extensively in engineering and social sciences. As an important branch of discrete mathematics, it raises general concern from all sides.Connectivity is one of the most basic graph characteristics, Connectivity is a powerful tool to analyze and describe graphs and a large number of problems can be attributed to the conditions for edge-connected graph problems, so this is a hot research field of graph theory. Currently, the Internet has been closely related to people's work, daily life. Connected graph has a strong application background because it has the combination of close contact with network optimization model. Connected Graph Edges and existence of contractible edges of graph structure, confirmed by the recursive nature of graphs, has important applications, thus it has important theoretical and practical value to study them. This paper can be selected in the contraction of a connected graph edge as the research object, the goal is to further the understanding of the structure of graphs and find out the constructor on future research help.This paper studies contractible edges in graphs properties and their specific sub-distribution map. The following explain the main results briefly.The first chapter of 6 - connected graph perfect matching on the distribution of contractible edges, the conclusions is as follow:Theorem Let G is a connected graph of order greater than 12,M is a perfect match, if the order of any fragments are greater than 3, then there are at least two contractible edges in G .ChapterⅡis on the basis of the first chapter, continue to explore the 6 - Connected Graph contractible edge, and get 6 - connected graph on the longest cycle in the distribution of contractible edges, the conclusions is following:Theorem Let G be a 6 - connected graph, and the order of any fragments are greater than 2 , C = x₁ x₂ ... x_nx₁is the longest of any circle, if one of the following conditions for any vertex_in C are met, then there are at least two contractible edges in G .(1) d ( x_i)≥7;(2) d ( x_i) = 6,then include it in acyclic.There is no non-contractible edges in a connected graph called complete contraction critical graphs. Connected graph of the critical contraction is currently a more popular topic, this paper contraction critical points connected graph degree distribution and the corresponding results in fragments.Theorem Let G be a contraction critical 6 - connected graph, x_is the arbitrary point ,set A is an atom, in mind, then there are 6 degrees with the adjacent points or the distance between two points is 2.Theorem Let G be a contraction critical 6 - connected graph, F is in fragments, and x∈N ( F). If |F|≥4,|(F|-)|≥3andN ( x )∩F = { x₁}, then there exists a point such that: x₂∈N ( F )∩N ( x )∩N ( x₁ )∩V₆ (G ).