Results Of Removable Edges In 5-Connected Graphs

The connectivity of graphs is one of the most important properties of graphs. Connected graphs plays an important role in practical applications because of its close connection with network model and combinatorial optimization. Removable edges are powerful tools to study the structure of graphs and to prove some properties of graphs by induction. They are very important in both theoretical respect and practical application . The main work of this paper is on removable edges in connected graphs in order to know more about the construction of connected graphs. This paper focus on the properties of removable edges in connected graphs, and their distributions in some special subgraphs of connected graphs. The following are the main results of this paper.First, give the definition of subgraphs of the graph:Definition: Let G be a 5-connected graph, if is a subgraph of G, and V(H) = {x, y, a, b, c, z, w}, E(H) = {xy, xa, xb, xc, xz, ab, ac, az, aw},d(x) = d(a) = 5, then H is called *1. By the same way, we give the definition of *2, *3, as follow:*2: V(H) = {x, y, a, b, c, z, w, u},E(H) = {xy, xa, xb, xc, xz, ab, ac, az, aw, by, bc, bu},d(x) = d(a) = d(b) = 5;*3: V(H) = {x, y, a, b, c, z, w, u, v, s},E(H) = {xy, xa, xb, xc, xz, za, zu, zc, zw},d(x) = d(z) = 5oConclusion 1. Let G be a 5-connected graph and |G|≥10, an edge e=xy of G is unremovable, its corresponding separating group is (e, S; A, B), and A is a 2-edge-vertex-cut atom in G, x∈A, y∈B. If there exists another unremovable edge f = xz(z≠y), its separating group is (f, T; C, D), x∈C, z∈D. Then there must be one of * 1, *2as a subgraph of G.Conclusion 2. Let G be a 5-connected graph and |G|≥10, C a cycle of G. If |E(C)∩E_R(G)| = 0, there is one of *1, *2, *3 as a subgraph of G.Conclusion 3. Let G be a 5-connected graph and there is no subgraph of G is * 1,*2, N(G)> is a tree, then |N(G)>|≤|G| - 4.Conclusion 4. Let G be a 5-connected graph, (?)≠E₀≠E_N(G), xy∈E₀ and its corresponding separating group is (xy, S; A, B), x∈A, y∈B, A is a E₀-edge-vertex-cut atom, then one conclusion below is corrected. (1)(E(A)∪(A,S))∩E₀ = (?);(2)(E(A)∪(A,S))∩E_R(G)≠(?).