Long Paths Through Specified Vertices In 3-Connected Graphs

Let h ≥ 6 be an integer, let G be a 3-connected graph with ∣V(G)∣ ≥ h − 1, and let x and z be distinct vertices of G. We show that if for any nonadjacent distinct vertices u and v in V(G) − {x, z}, the sum of the degrees of u and v in G is greater than or equal to h, then for any subset Y of V(G) − {x, z} with ∣Y∣ ≤ 2, G contains a path which has x and z as its endvertices, passes through all vertices in Y, and has length at least h − 2. We also show a similar result for cycles in 2-connected graphs.     

On a Spanning K-tree Containing Specified Vertices in a Graph

Acta Mathematicae Applicatae Sinica, English Series - A k-tree is a tree with maximum degree at most k. In this paper, we give a sharp degree sum condition for a graph to have a spanning k-tree in...     

收缩临界6连通图的6度顶点

利用断片的性质,改进了齐恩凤,齐登记等的研究结果,得到了收缩临界6-连通图中6度点的性质的新结果:设x是G中任意一点,设A是一个x-原子,记N_A=T_A,N(x)∩T_A≠Φ,则A∩T_A中有与x相邻的6度点或两点的距离为2.     

Long Cycles Passing Through a Specified Edge in 3-Connected Graphs

 In this paper we prove that if G is a 3-connected noncomplete graph of order n satisfying that the degree sum of any two vertices with distance 2 is not less than m, then either there exists a cycle containing e of length at least min{n,m} for any edge e of G, or

A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph

For a graph G, we define σ₂(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k of length at most four such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k. And show if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k, V(C ₁) ∪...∪ V(C _k ) = V(G), and |C _i | ≤ 4 for all 1 ≤ i ≤ k − 1. The condition of degree sum σ₂(G) ≥ n + k − 1 is sharp. Received: December 20, 2006. Final version received: December 12, 2007.     

On the Number of Vertices with Specified Eccentricity

The eccentricity of a vertex v in a graph is the maximum of the distances from v to all other vertices. The diameter of a graph is the maximum of the eccentricities of its vertices. Fix the parameters n, d, c. Over all graphs with order n and diameter d, we determine the maximum (within 1) and the minimum of the number of vertices with eccentricity c. Revised: May 7, 1999     

The Graph of Triangulations of a Point Configuration with d +4 Vertices Is 3-Connected

We study the graph of bistellar flips between triangulations of a vector configuration A with d+4 elements in rank d+1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d+4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and, in the acyclic case, every triangulation has at least three geometric bistellar neighbours. In corank 4, connectivity is not known and having at least four flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based on a dualization of the concept of triangulation of a point or vector configuration A to that of a virtual chamber of its Gale transform B , introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector configuration B can be realized as a cell in some pseudo-chamber complex of B in the same way that regular triangulations appear as cells in the usual chamber complex. All the results in this paper generalize to triangulations of corank 3 oriented matroids and virtual chambers of rank 3 oriented matroids, realizable or not. The details for this generalization are given in the Appendix. Received March 1, 1999, and in revised form September 7, 1999.     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

On the Existence of Disjoint Cycles in a Graph

G is a graph of order at least 3k with . Then G contains k vertex-disjoint cycles. Received: April 23, 1998     

Long Snakes in Powers of the Complete Graph with an Odd Number of Vertices

In [5] Abbott and Katchalski ask if there exists a constantc < 0 such that for every d 2 there is a snake (cycle withoutchords) of length at least c3^d in the product of d copies ofthe complete graph K₃. We show that the answer to the abovequestion is positive, and that in general for any odd integern there is a constant c_n such that for every d 2 there is asnake of length at least c_n n^d in the product of d copies ofthe complete graph K_n.     

On the Number of Subgraphs of a Specified Form Embedded in a Random Graph

Let U ₁, U ₂,... be a sequence of i.i.d. random elements in R^d. For x>0, a graph G _n (x) may be formed by connecting with an edge each pair of points in that are separated by a distance no greater than x. The points of G _n (x) could represent the stations in a telecommunications network and the edge set the lines of communication that exist among them. Let be a collection of graphs on mn points having a specified form or structure, and let denote the number of subgraphs embedded in G _n (x) and contained in . It is shown that a SLLN, CLT and LIL for follow easily from the theory of U-statistics. In addition, a uniform (in x) SLLN is proved for collections that satisfy a certain monotonicity condition. Some applications are mentioned and the results of some simulations presented. The scaling constants appearing in the CLT are usually hard to obtain. These are worked out for some special cases.     

Every 4-Connected Line Graph of a Planar Graph is Hamiltonian

Let G be a graph withE(G) $#x2260;ø. The line graph of G, written L(G) hasE(G) as its vertex set, where two vertices are adjacent in L(G) if and only if the corresponding edges are adjacent inG. Thomassen conjectured that all 4-connected line graphs are hamiltonian [2]. We show that this conjecture holds for planar graphs.     

The Orders of Vertices of a Character-degree Graph

Let G be a finite group. We put ρ(G) = {p|p is a prime dividing χ(1) for some χ ∈Irr(G)}. We define a graph Γ(G), whose vertices are the primes in ρ(G) and p, q ∈ ρ(G) are connected in Γ(G) denoted p ～ q, if pq||χ(1) for some χ ∈Irr(G). For p ∈ ρ(G), we define ord(p) = |{q ∈ ρ(G)|q ～ p}|. We call ord(p) the order of the vertex p of the graph Γ(G). In this article, we discuss orders and the influences of orders on the structure of finite groups.     

一个六阶3-连通图与路Pn的笛卡尔积的交叉数

C(6,2)表示由圈C6增加边vivi 2(i=1,…,6,i 2(m od6))所得的图,把边vivi 2叫做C(6,2)的弦,B表示C(6,2)除去一条弦所得到的图,我们确定了B与Pn笛卡尔积的交叉数为5n-1.     

Coverings of the Vertices of a Graph by Small Cycles

Given a graph G with n vertices, we call c_k(G) the minimum number of elementary cycles of length at most k necessary to cover the vertices of G. We bound c_k(G) from the minimum degree and the order of the graph.     

On Partitioning the Edge Set of a Graph into Internally Disjoint Paths without Exterior Vertices

A collection of nontrivial paths in a graph G is called a path pile of G, if every edge of G is on exactly one path and no two paths have a common internal vertex. The least number that can be the cardinality of a path pile of G is called the path piling number of G. It can be shown that ε ≤ ν + η where ε, ν and η are respectively the size, the order and the path piling number of G. In this note we characterize structurally the class of all graphs for which the equality of this relation holds.