Some Results Of Cyclic Structure In Graphs And Weighted Graphs

In this paper, we disscuss the problem on cyclic structure of graphs . We provide three sufficient conditions on the existence of long cycles passing through a specified vertex, a specified edge in unweighted graphs, and three sufficient conditions on the existence of heavy cycles in weighted graph. The results are as follows:Theorem 1 Let G be a 3-connected graph of order n and s an integer (3< s < n).For any u V(G), if d(u)=k < s/2 implies (where then through each vertex of G there passes a cycle of length >s.Theorem 2 Let G is a 3-connected graph and s an integer (3 < s < n). For each induced subgraph L of G isomorphic to K1,3 or M1(where M1 denotes K1,3+ e, e an edge), if d(z) < s/2 and dL(x,y)=2(where x,y V(L)) imply d(y) > s/2. Then through each vertex of G there passes a cycle of length > s. is also the degree of vertice in graph and satisfies: (I). is as great as possible,Theorem 3 Let G is a 2-connected A-free graph. Then through each edge of G there passes a cycle of length > min|V(G)|,2maxTheorem 4 Let G is a 2-connected nonhamilton weighted graph satisfying for any independent set S={u,v,w}, there exist x=y S such that DW(x)+dW(y)>m, Then cw(G)>m.In Theorem 5 and Theorem 6 Conditions D1 and D2 are defined as follows:(D2). In every triangle of graph G, either all edges have different weights or all edges have same weights .Theorem 5 Let G is a 2-connected nonhamilton weighted graph satisfying conditions D1 and D2. For each induced subgraph L of G isomorphic to K1,3 or M1 (where M1 denotes K1,3+e, e an edge), if dw(x) < m/2 and dL(x,y) = 2(where x,y 6 V(L)) imply m/2, then cw(G)>m.Theorem 6 Let G is a 2-connected nonhamilton weighted graph satisfying conditions D1 and D2. If dw(x)+dw(y)>m(xy E(G), x=y), then through each vertex of G there passes a cycle of weight at least m.