K-connected Research Articles

On Frank's conjecture on k-connected orientations

We disprove a conjecture of Frank [3] stating that each weakly 2k-connected graph has a k-vertex-connected orientation. For k≥3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

Bipartitions of highly connected tournaments

We show that if T is a strongly 109k6log⁡(2k)-connected tournament, there exists a partition A, B of V(T) such that each of T[A], T[B] and T[A,B] is strongly k-connected. This provides tournament analogues of two partition conjectures of Thomassen regarding highly connected graphs.

Removable edges in cycles of a k-connected graph

Science-technology Foundation for Young Scientists of Fujian Province [2007F3070]; National Natural Science Foundation of China [10831001]

On minimally k-connected matroids

A matroid M is minimally k-connected if M is k-connected and, for every e ∈ E ( M ) , M \\ e is not k-connected. It is conjectured that every minimally k-connected matroid with at least 2 ( k − 1 ) elements has a cocircuit of size k. We resolve the conjecture almost affirmatively for the case k = 4 by finding the unique counterexample; and for each k ⩾ 5 , we prove that there exists a counterexample to the conjecture with 2 k + 1 elements.

Proof of Mader's conjecture on k-critical n-connected graphs

Mader conjectured that every k-critical n-connected noncomplete graph G has 2k+2 pairwise disjoint fragments. The author in [9] proved that the conjecture holds if the order of G is greater than (k...

Closure and stable Hamiltonian properties in claw-free graphs

In the class of k-connected claw-free graphs, we study the stability of some Hamiltonian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an infinite family of graphs Gk of arbitrarily high connectivity k such that the closure of Gk has the property while the graph Gk does not); (ii) traceability is a stable property even for k = 1; (iii) homogeneous traceability is not stable for k = 2 (although it is stable for k = 7). The article is concluded with several open questions concerning stability of homogeneous traceability and Hamiltonian connectedness. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 30–41, 2000

Dλ-cycles in λ-claw-free graphs

For a given integer λ ⩾ 1, we define a λ-claw as a graph H which is connected and contains a vertex w such that H- w is the union of three pairwise remote connected subgraphs with exactly λ vertices each. We define a graph to be λ- claw- free if it contains no induced subgraph isomorphic to a λ-claw. We show that if G is a k-connected ( k integer, k ⩾ 2), λ-claw-free graph such that if for every two remote connected subgraphs S 1, S 2 of G of order λ we have d( S 1) + d( S 2) > n − ( k + 1) λ − k, or if for every three pairwise remote, connected subgraphs S 1, S 2, S 3 of G of order λ we have d( S 1) + d( S 2) + d( S 3) > n − ( k + 1) λ + 1, then G contains a cycle C such that every component of G − V( C) has at most λ − 1 vertices.

On Immersions of k-Connected n-Manifolds

On Immersions of k-Connected n-Manifolds

Connectivity and Hamiltonian Connectedness of Graphs

1 Introduction In this note we consider simple graphs only, following Bondy and Murty for basic graph-theoretic notations and terminology. In addition, we set G_n~k={G|G is a k-connected graph of order n}, H_c={G|G is a Hamiltonian connected graph}, and denote by P_H(u,v)a Hamiltonian u-v path. A path P in a graph G is called a dominating path if every edge of G has at least one of its end vertices on P. α and K denote the independence number and connectivity of G.

Cycles through k+2 vertices in k-connected graphs

Soit G un graphe k-connexe dans lequel les (k+1) sommets sont cocycliques. Si k≥4, alors les 2 conditions suivantes sont equivalentes: 1) (k+2) sommets ne sont pas cycliques; 2) a) il y a des sommets s 1 ,s 2 ,...,s k dans G, dont le retrait produit plus de (k+1) composantes; b) Il y a des ensembles V o ,V 1 ,...,V k+2 de sommets dans G et des sommets distincts x 1 ,x 2 ,x 3 ,s 1 ,s 2 ,...s k tels que:V o ∩V j ={x j ,s 1 ,s 2 ,...,s k } quand 1≤j≤3, V o ∩V j ={s 1 ,s 2 ,...,s k } quand 4≤j≤k+2, V i ∩V j ={s 1 ,s 2 ,...,s k } quand 1≤i<j≤k+2, V ji¬=j ∪V i ¬=0 quand 1≤j≤k+2 et tels que chaque arete de G a 2 bouts dans un Vi (avec o≤i≤k+2)

On approximate n-connectedness

On approximate n-connectedness

Spanning subgraphs of k-connected digraphs

It is shown that every k-edge-connected digraph with m edges and n vertices contains a spanning connected subgraph having at most 2m + 6(k −1)(n − 1)) (5k − 3) edges. When k = 2 the bound is improved to (3m + 8(n − 1)) 10 , which implies that a 2-edge-connected digraph is connected by less than 70% of its edges. Examples are given which require almost two-thirds of the edges to connect all vertices.

On certain n-connected matroids.

On certain n-connected matroids.

Knotting a k-Connected Closed PL m-Manifold in E 2m-k

Knotting a k-Connected Closed PL m-Manifold in E 2m-k

Characterizations of n-connected and n-line connected graphs

The following two theorems are proved: (1) A graph G with at least n + 1 points, n ≥ 2, is n-connected if and only if for each set S of n distinct points of G and for each two point subset T of S there is a cycle of G that contains the points of T and avoids the points of S − T. (2) A graph G with at least n + 1 lines, n ≥ 2, with no isolated points is n-line connected if and only if for each set S of n distinct lines of G and for each two line subset T of S there is a circuit of G that contains the lines of T and avoids the lines of S − T.