The Contractible Edges And Removable Edges Of The Connected Graph

The connectivity of the graph is an important part of the graph theory, so the study of the structure of the connected graph has always been one of the impor-tant subjects of the graph theory.The existence of contractible and removable of connected graphs plays an important role in studying the structure of connected graphs. In this paper, we discuss the distributions of the contractible edges of 5-connected graphs and k-connected graphs on the longest circle, spanning tree,perfect matching, and the properties of the removable edges. The profiles of three main content are described in detail in following:First of all, using the atoms, the properties of the fragment that does not contain a special 2-fragment of 5-connected graph of the longest circle C has at least 6 contractible edges, It is further proved that if there is no 3 cycles in C that contains 5 degrees,, C has at least 2 contractible edges; We prove that there is at least 6 contractible edges of the 5-connected graph spanning tree that does not contain special 2-fragment, and get a class of 5-connected graphs Hn of E(H) (?) En(G); We Give a 5-connected graph that does not contain a special 2-fragment if the perfect match is not within 3 cycles, Perfect match M has at least 8 contractible edges, further we show that if g(G) > 4, There are at least 8 contractible edges on the M.Secondly, we study the distribution of the contractible edges of ?-connected graphs, and prove that if {x} (?)C T is cut and E(x) (?) En(G), then there is a point y, d(y)= ?,y?N(F) ? N(x). We Also give the A - connected graph of the longest circle, the spanning tree and the perfect match on the contractible edges of the more general lower bound.Finally, we study the properties of the ?-connected graph, and we show that there are at least two removable edges on the spanning tree H of ?-connected graph to satisfy the least degree. The order of the atoms in the ?-connected graph is at least ? - 3, The edges on G[A] and G[S] are both removable.