N-connected Graph Research Articles

Matching Connectivity: On the Structure of Graphs with Perfect Matchings

We introduce the concept of matching connectivity as a notion of connectivity in graphs admitting perfect matchings. The notion relies heavily on structural properties of those matchings. We prove a Menger-type result for matching n-connected graphs. Furthermore, we show that matching connectivity fills a gap in the investigation of n-extendable graphs and their connectivity properties. In particular, we show that every n-extendable graph is matching n-connected and for the converse any matching (n + 1)-connected graph either is n-extendable, or belongs to a well described class of graphs: the brace h-critical graphs.

Formula omitted]-dominating [formula omitted]-trees of graphs

Let k≥2, l≥2, m≥0 and n≥1 be integers, and let G be a connected graph. If there exists a subgraph H of G such that for every vertex v of G, the distance between v and H is at most m, then we say that H m-dominates G. A tree whose maximum degree is at most k is called a k-tree. Define αl(G)=max{|S|:S⊆V(G),dG(x,y)≥lfor all distinctx,y∈S}, where dG(x,y) denotes the distance between x and y in G. We prove the following theorem and show that the condition is sharp. If an n-connected graph G satisfies α2(m+1)(G)≤(k−1)n+1, then G has a k-tree that m-dominates G. This theorem is a generalization of both a theorem of Neumann-Lara and Rivera-Campo on a spanning k-tree in an n-connected graph and a theorem of Broersma on an m-dominating path in an n-connected graph.

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node $$r$$r and for any other node $$v(\ne r)$$v(?r), the paths from $$v$$v to $$r$$r in any two trees are node-disjoint except the two end nodes $$v$$v and $$r$$r. It was conjectured that for any $$n$$n-connected graph there exist $$n$$n ISTs rooted at an arbitrary node. Let $$N=2^n$$N=2n be the number of nodes in the $$n$$n-dimensional Mobius cube $$MQ_n$$MQn. Recently, for constructing $$n$$n ISTs rooted at an arbitrary node of $$MQ_n$$MQn, Cheng et al. (Comput J 56(11):1347---1362, 2013) and (J Supercomput 65(3):1279---1301, 2013), respectively, proposed a sequential algorithm to run in $${\mathcal O}(N\log N)$$O(NlogN) time and a parallel algorithm that takes $${\mathcal O}(N)$$O(N) time using $$\log N$$logN processors. However, the former algorithm is executed in a recursive fashion and thus is hard to be parallelized. Although the latter algorithm can simultaneously construct $$n$$n ISTs, it is not fully parallelized for the construction of each spanning tree. In this paper, we present a non-recursive and fully parallelized approach to construct $$n$$n ISTs rooted at an arbitrary node of $$MQ_n$$MQn in $${\mathcal O}(\log N)$$O(logN) time using $$N$$N nodes of $$MQ_n$$MQn as processors. In particular, we derive useful properties from the description of paths in ISTs, which make the proof of independency to become easier than ever before.

A comment on “Independent spanning trees in crossed cubes”

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node r and for any other node v(≠r), the paths from v to r in any two trees are node-disjoint except the two end nodes v and r. For an n-connected graph, the independent spanning trees problem asks to construct n ISTs rooted at an arbitrary node of the graph. Recently, Zhang et al. (2013) [18] proposed an algorithm to construct n ISTs with a common root at node 0 in an n-dimensional crossed cube CQn. However, it has been proved by Kulasinghe and Bettayeb (1995) [13] that the CQn (a synonym called multiply-twisted hypercube in that paper) fails to be node-transitive for n⩾5. Thus, the result of Zhang et al. does not really solve the ISTs problem in CQn. In this paper, we revisit the problem of constructing n ISTs rooted at an arbitrary node in CQn. As a consequence, we show that the proposed algorithm can be parallelized to run in O(log⁡N) time using N=2n nodes of CQn as processors.

Independent spanning trees in crossed cubes

Multiple independent spanning trees (ISTs) can be used for data broadcasting in networks, which can provide advantageous performances, such as the enhancement of fault-tolerance, bandwidth, and security. However, there is a conjecture on the existence of ISTs in graphs: If a graph G is n-connected (n⩾1), then there are n ISTs rooted at an arbitrary vertex in G. This conjecture has remained open for n⩾5. The n-dimensional crossed cube CQn is a n-connected graph with various desirable properties, which is an important variant of the n-dimensional hypercube. In this paper, we study the existence and construction of ISTs in crossed cubes. We first give a proof of the existence of n ISTs rooted at an arbitrary vertex in CQn(n⩾1). Then, we propose an O(Nlog2N) constructive algorithm, where N=2n is the number of vertices in CQn.

Independent Spanning Trees on Special BC Networks

There is a well-known conjecture on independent spanning trees (ISTs) on graphs: For any n-connected graph G with n≥1, there are n ISTs rooted at an arbitrary node on G. It still remains open for n≥5. We propose an integrated algorithm to construct n ISTs rooted at any node similar to 0 or 10n-1 on n-dimensional HCH cube for n≥1 and give the simulations of ISTs on several special BC networks, such as HCH cubes, crossed cubes, Möbius cubes, twisted cubes, etc.

An algorithm to construct independent spanning trees on parity cubes

Independent spanning trees have applications in networks such as reliable communication protocols, one-to-all broadcasting, reliable broadcasting, and secure message distribution. Thus, the designs of independent spanning trees in several classes of networks have been widely investigated. However, there is a conjecture on independent spanning trees: any n-connected graph has n independent spanning trees rooted at an arbitrary vertex. This conjecture still remains open for n≥5. In this paper, by proposing an algorithm to construct n independent spanning trees rooted at any vertex, we confirm the conjecture on n-dimensional parity cube PQn —— a variant of n-dimensional hypercube. Furthermore, we prove that all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n+1 for any integer n≥2.

Spanning k-Trees of n-Connected Graphs

A tree is called a k-tree if the maximum degree is at most k. We prove the following theorem, by which a closure concept for spanning k-trees of n-connected graphs can be defined. Let k ≥ 2 and n ≥ 1 be integers, and let u and v be a pair of nonadjacent vertices of an n-connected graph G such that deg G (u) + deg G (v) ≥ |G| − 1 − (k − 2)n, where |G| denotes the order of G. Then G has a spanning k-tree if and only if G + uv has a spanning k-tree.

Degree Bounded Spanning Trees

In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set $${S \subseteq V(G)}$$of cardinality n(k−1) + c + 2, there exists a vertex set $${X \subseteq S}$$of cardinality k such that the degree sum of vertices in X is at least |V(G)| − c −1. Then G has a spanning tree T with maximum degree at most $${k+\lceil c/n\rceil}$$and $${\sum_{v\in V(T)}\max\{d_T(v)-k,0\}\leq c}$$.

Cycle-minors and subdivisions of wheels

In this paper we prove two results. The first is an extension of a result of Dirac which says that any set of n vertices of an n-connected graph lies in a cycle. We prove that if V′ is a set of at most 2n vertices in an n-connected graph G, then G has, as a minor, a cycle using all of the vertices of V′. The second result says that if G is an n+1-connected graph with maximum vertex degree Δ then G contains a subgraph that is a subdivision of W2n if and only if Δ≥2n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 100–108, 2009

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.

Connectivity keeping edges in graphs with large minimum degree

The old well-known result of Chartrand, Kaugars and Lick says that every k-connected graph G with minimum degree at least 3 k / 2 has a vertex v such that G − v is still k-connected. In this paper, we consider a generalization of the above result [G. Chartrand, A. Kaigars, D.R. Lick, Critically n-connected graphs, Proc. Amer. Math. Soc. 32 (1972) 63–68]. We prove the following result: Suppose G is a k-connected graph with minimum degree at least ⌊ 3 k / 2 ⌋ + 2 . Then G has an edge e such that G − V ( e ) is still k-connected. The bound on the minimum degree is essentially best possible.

Circuits and Cocircuits in Regular Matroids

A classical result of Dirac's shows that, for any two edges and any n−2 vertices in a simple n-connected graph, there is a cycle that contains both edges and all n−2 of the vertices. Oxley has asked whether, for any two elements and any n−2 cocircuits in an n-connected matroid, there is a circuit that contains both elements and that has a non-empty intersection with all n−2 of the cocircuits. By using Seymour's decomposition theorem and results of Oxley and Denley and Wu, we prove that a slightly stronger property holds for regular matroids.

The super connectivity of shuffle-cubes

The shuffle-cube SQ n , where n ≡ 2 ( mod 4 ) , a new variation of hypercubes proposed by Li et al. [T.-K. Li, J.J.M. Tan, L.-H. Hsu, T.-Y. Sung, The shuffle-cubes and their generalization, Inform. Process. Lett. 77 (2001) 35–41], is an n-regular n-connected graph. This paper determines that the super connectivity of SQ n is 2 n − 4 and the super edge-connectivity is 2 n − 2 for n ⩾ 6 .

High Connectivity Keeping Sets In n-Connected Graphs

It is proved that for every positive integer k, every n-connected graph G of sufficiently large order contains a set W of k vertices such that G—W is (n-2)-connected. It is shown that this does not remain true if we add the condition that G(W) is connected.

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...

Cycles through a prescribed vertex set in N-connected graphs

A well-known result of Dirac (Math. Nachr. 22 (1960) 61) says that given n vertices in an n-connected G, G has a cycle through all of them. In this paper, we generalize Dirac's result as follows: Given at most 3 2 n vertices in an n-connected graph G when n⩾3 and |V(G)|⩾ 3 2 n , then G has a cycle through exactly n vertices of them. This improves the previous known bound given by Kaneko and Saito (J. Graph Theory 15(6) (1991) 655).

On k-con-Critically n-Connected Graphs

We prove that every n-connected graph G of sufficiently large order contains a connected graph H on four vertices such that G− V( H) is ( n−3)-connected. This had been conjectured in Mader (High connectivity keeping sets in n-connected graphs, Combinatorica, to appear). Furthermore, we prove upper bounds for the order of all n-connected graphs of criticality 3, 4, and 5.

The shuffle-cubes and their generalization

In this paper, we first present a new variation of hypercubes, denoted by SQ n . SQ n is obtained from Q n by changing some links. SQ n is also an n-regular n-connected graph but of diameter about n/4. Then, we present a generalization of SQ n . For any positive integer g, we can construct an n-dimensional generalized shuffle-cube with 2 n vertices which is n-regular and n-connected. However its diameter can be about n/g if we consider g as a constant.

On locally k-critically n-connected graphs

Let ∅ ≠ W ⊆ V( G). The graph G is called a W-locally k-critically n-connected graph or simply a W-locally ( n, k)-graph, if for all V′ ⊆ W with | V′|⩽ k and each fragment F of G we have that κ( G− V′)= n−| V′| and F ∩ W ≠ ∅. In this paper we prove that every non-complete W-locally ( n, k)-graph has (2 k+2) distinct fragments and | W|⩾2 k+2. From this result it follows that: (1) Let G be a non-complete ( n, k)-graph. If all ends of G are proper, then G has (2 k+2) pairwise disjoint ends. (2) Slater's conjecture on ( n, k)-graphs holds, i.e., the complete graph K n+1 is the unique ( n, k)-graph for 2 k> n.