Figure About Some Of The Results Of The X-circle (road)

Let G=(V, E) be a 3-connected graph on n vertices and X(?)V(G). A cycle C ofG is X-longest if |X∩V(C)|≥|X∩V(C′)|for every cycle C' of G. Byα(G) wedenote the independence number of G, byα(X) the independence number of G[X]and byσ_k(X) the minimum degree sum (in G)of any k pairwise nonadjacent verticesof X when k≤α(X), for k＞α(X), we setσ_k(X)=k(n-α(X)). A path P of G iscalled a X-path if X(?)V(P). (n₁, n₂,…, n_k) is called a k-partition of n if n=∑_i=1^k n_iand n_i≥2(1≤i≤k), where n_i(1≤i≤k) are all integers.In [4], Yuan Zou gives the following conclusion: G is a 3-connected graph on nvertices, C is a longest cycle of G, R=G\C. Ifσ₄(G)≥n+6, then for every componentH of G[R], |V(H)|≤2. We changeσ₄(G) in the condition, that is, let X(?)V(G),we changeσ₄(G) toσ₄(X), then we can get an extension of the conclusion of YuanZou: G is a 3-connected graph on n vertices, X(?)V(G), C is a X-longest cycle of G,R=G\C. Ifσ₄(X)≥n+6, then for every component H of G[R], |V(H)∩X|≤2and every component which contains two vertices in X can not contains vertices inV(G)\X. This is the first conclusion of this paper.In [8], Enomoto gives that G is a connected graph on n vertices. Ifσ₃(G)≥norα(G)≤2, then G has a hamilton path or every longest cycle of G is stronglydominating. We improve part of this conclusion, then we get the second conclusionof this paper: G is a connected graph on n vertices, X(?)V(G). Ifα(X)≤2, then Ghas a X-path.In [6], Yaojun Chen gives that G is a connected graph on n vertices, ifσ₃(G)≥(3n-5)/2, then G has a hamilton path. we can improve this conclusion to get the thirdconclusion: G is a connected graph on n vertices, X(?)V(G). Ifσ₃(X)≥(3n-5)/2,then G has a X- path. Obviously, this conclusion is an extension of the conclusion ofYaojun Chert.Similarly, in [7],Robert Jahansson gives that let G be a connected graph with nvertices and (n₁, n₂,…, n_k)is a k-partition of n, where all n_i are odd positive integers.Suppose furthermore that G contains a path P such that every vertex v∈V(G)\V(P) hasno two consecutive neighbors on P and satisfies d_P(v)≥(?)n₁/2」+(?)n₂/2」+…+ (?)n_k/2」, then G contains vertex disjoint paths of orders n₁,…, n_k. We can improve thisconclusion to get the fourth conclusion: G is a connected graph with n vertices, X(?)V(G), |X|=s, (s₁, s₂,…, s_k)is a k-partition of s, where all s_i are odd positive integers.Suppose furthermore that G contains a path P such that every vertex x∈X\V(P)has no two consecutive neighbors in X on P and satisfies |N_P(v)∩X|≥(?)s₁/2」+(?)s₂/2」+…+(?)s_k/2」, then G contains vertex disjoint paths P₁, P₂,…, P_k such that|P_i∩X|=s_i(i=1, 2,…, k). We can see the conclusion of Robert Jahansson is acorollary of this conclusion.In the last chapter, we give some problems for deeper study.