Cyclic Edge-connectivity Research Articles

Symmetric Property and Reliability of Balanced Hypercube

Huang and Wu in [IEEE Transactions on Computers 46 (1997) 484-490] introduced the balanced hypercube $BH_n$ as an interconnection network topology for computing systems, and they proved that $BH_n$ is vertex-transitive. However, some other symmetric properties, say edge-transitivity and arc-transitivity, of $BH_n$ remained unknown. In this paper, we solve this problem and prove that $BH_n$ is an arc-transitive Cayley graph. Using this, we also investigate some reliability measures, including super-connectivity, cyclic connectivity, etc., in $BH_n$ . First, we prove that every minimum edge-cut of $BH_n (n\ge 2)$ isolates a vertex, and every minimum vertex-cut of $BH_n (n\ge 3)$ isolates a vertex. This is stronger than that obtained by Wu and Huang which shows the connectivity and edge-connectivity of $BH_n$ are $2n$ . Second, Yang [Applied Mathematics and Computation 219 (2012) 970-975.] proved that for $n\ge 2$ , the super-connectivity of $BH_n$ is $4n-4$ and the super edge-connectivity of $BH_n$ is $4n-2$ . In this paper, we proved that $BH_n (n\ge 2)$ is super- $\lambda^{\prime }$ but not super- $\kappa^{\prime }$ . That is, every minimum super edge-cut of $BH_n (n\ge 2)$ isolates an edge, but the minimum super vertex-cut of $BH_n (n\ge 2)$ does not isolate an edge. Third, we also obtain that for $n\ge 2$ , the cyclic connectivity of $BH_n$ is $4n-4$ and the cyclic edge-connectivity of $BH_n$ is $4(2n-2)$ . That is, to become a disconnected graph which has at least two components containing cycles, we need to remove at least $4n-4$ vertices (resp. $4(4n-2)$ edges) from $BH_n (n\ge 2)$ .

Atoms of cyclic edge connectivity in regular graphs

A cyclic edge-cut of a connected graph $$G$$G is an edge set, the removal of which separates two cycles. If $$G$$G has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph $$G$$G, the cyclic edge connectivity of a graph $$G$$G, denoted by $$\lambda _c(G)$$?c(G), is the minimum cardinality over all cyclic edge cuts. Let $$X$$X be a non-empty proper subset of $$V(G)$$V(G). If $$[X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}$$[X,X¯]={xy?E(G)|x?X,y?X¯} is a minimum cyclic edge cut of $$G$$G, then $$X$$X is called a $$\lambda _c$$?c-fragment of $$G$$G. A $$\lambda _c$$?c-fragment with minimum cardinality is called a $$\lambda _c$$?c-atom. Let $$G$$G be a $$k (k\ge 3)$$k(k?3)-regular cyclically separable graph with $$\lambda _c(G)

On forcing matching number of boron–nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M . A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G . The smallest cardinality of forcing sets of M is called the forcing number of M . Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron–nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.

Super cyclically edge connected transitive graphs

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity ? c (G) is the cardinality of a minimum cyclic edge-cut of G. We call a graph super cyclically edge-connected, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In this paper, we show that a connected vertex-transitive or edge-transitive graph is super cyclically edge-connected if either G is cubic with girth g(G)?7, or G has minimum degree ?(G)?4 and girth g(G)?6.

Forcing matching numbers of fullerene graphs

The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G . In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.

On cyclic edge-connectivity of transitive graphs

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is said to be cyclically separable. For a cyclically separable graph G , the cyclic edge-connectivity c λ ( G ) is the cardinality of a minimum cyclic edge-cut of G . In this paper, we first prove that for any cyclically separable graph G , c λ ( G ) ≤ ζ ( G ) = min { ω ( X ) ∣ X induces a shortest cycle in G } , where ω ( X ) is the number of edges with one end in X and the other end in V ( G ) ∖ X . A cyclically separable graph G with c λ ( G ) = ζ ( G ) is said to be cyclically optimal. The main results in this paper are: any connected k -regular vertex-transitive graph with k ≥ 4 and girth at least 5 is cyclically optimal; any connected edge-transitive graph with minimum degree at least 4 and order at least 6 is cyclically optimal.

Formula omitted]-Optimally half vertex transitive graphs with regularity k

The cyclic edge connectivity λ c ( G ) of a graph G is the minimum number of edges whose removal results in a disconnected graph and every connected component contains cycles. If λ c ( G ) exists, we call G λ c -connected. And if λ c ( G ) gets the maximum value, we call G λ c -optimal. A bipartite graph is said to be half vertex transitive if its automorphism group is transitive on the sets of its bipartition. In this paper, we show that a connected k ( ⩾ 4 ) -regular half vertex transitive graph G with girth g ⩾ 6 is λ c -optimal, and we also obtain a sufficient and necessary condition for a connected k ( ⩾ 4 ) -regular half vertex transitive graph G with girth 4 to be λ c -optimal.

On cyclic edge-connectivity of fullerenes

A graph is said to be cyclically k-edge-connected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclic-k-edge cutset and it is called a trivial cyclic-k-edge cutset if at least one of the resulting two components induces a single k-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclically 5-edge-connected. In this article it is shown that a fullerene F containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that F has a Hamilton cycle, and as a consequence at least 15 · 2 n / 20 - 1 / 2 perfect matchings, where n is the order of F.

Characterization of graphs with infinite cyclic edge connectivity

In this paper, we characterize the graphs with infinite cyclic edge connectivity. Then we design an efficient algorithm to determine whether a graph has finite cyclic edge connectivity or infinite cyclic edge connectivity.

A survey on snarks and new results: Products, reducibility and a computer search

In this paper we survey recent results and problems of both theoretical and algorithmic character on the construction of snarks—non-trivial cubic graphs of class two, of cyclic edge-connectivity at least 4 and with girth ≥ 5. We next study the process, also considered by Cameron, Chetwynd, Watkins, Isaacs, Nedela, and Sˇkoviera, of splitting a snark into smaller snarks which compose it. This motivates an attempt to classify snarks by recognizing irreducible and prime snarks and proving that all snarks can be constructed from them. As a consequence of these splitting operations, it follows that any snark (other than the Petersen graph) of order ≤ 26 can be built as either a dot product or a square product of two smaller snarks. Using a new computer algorithm we have confirmed the computations of Brinkmann and Steffen on the classification of all snarks of order less than 30. Our results recover the well-known classification of snarks of order not exceeding 22. Finally, we prove that any snark G of order ≤ 26 is almost Hamiltonian, in the sense that G has at least one vertex v for which G \ v is Hamiltonian. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 57–86, 1998

Connectivity and β-invariants of isotropic systems and 4-regular graphs

For every matroid M, H. Crapo defined an invariant β( M) which reflects some connectivity properties of M. We adapt the definition of this invariant to isotropic systems. As an application we study the properties of an invariant β 4( H) defined by P. Martin for any 4-regular graph H, and we show that β 4( H) is related with the cyclic edge-connectivity of H.

N‐extendability of symmetric graphs

AbstractIt is proved that a cyclically (k − 1)(2n − 1)‐edge‐connected edge transitive k‐regular graph with even order is n‐extendable, where k ≥ 3 and k − 1 ≥ n ≥ ⌈(k + 1)/2⌉. The bound of cyclic edge connectivity is sharp when k = 3. © 1993 John Wiley & Sons, Inc.

Lower bound of cyclic edge connectivity for n-extendability of regular graphs

A cyclically m-edge-connected n-connected k-regular graph is called an ( m. n. k) graph. It is proved that for any m > 0 and k ⩾3, there is an ( m, k, k) bipartite graph. A graph G is n-extendable if every matching of size n in G lies in a perfect matching of G. We prove the existence of a ( k 2 −1, k + 1, k + 1) bipartite graph which is not k-extendable and the existence of an ( m, k + 1, k + 1) graph which is not n-extendable, where n ⩾ 2, k ⩾ 2 and m is any positive integer. The existence of the former graphs shows that a result of Holton and Plummer is sharp.

Three‐regular subgraphs of four‐regular graphs

AbstractBerge conjectured that every finite simple 4‐regular graph G contains a 3‐regular subgraph. We prove that this conjecture is true if the cyclic edge connectivity λc(G) of G is at least 10. Also we prove that if G is a smallest counterexample, then λc(G) is either 6 or 8.