Cyclic Edge-connectivity Research Articles

Super (Edge-)connectivity of the Folded Petersen Network

Reliability evaluation of an interconnection network is of great significance for construction and maintenance of the network. The super (edge-)connectivity, restricted (edge-)connectivity and cyclic (edge-)connectivity are important parameters to evaluate network reliability. Compared with the hypercube [Formula: see text], the folded Petersen network [Formula: see text] has better properties, including strong connectivity, symmetry. Besides, it has smaller diameter and more vertices than [Formula: see text] with the same degree and connectivity. In this paper, based on the symmetric properties of [Formula: see text], we prove that [Formula: see text] is super (edge-)connected, super restricted edge-connected and super cyclically edge-connected. In addition, we obtain the restricted edge-connectivity and the cyclic edge-connectivity of [Formula: see text].

Cyclic edge and cyclic vertex connectivity of [formula omitted]-fullerene graphs

Cyclic vertex connectivity cκ and cyclic edge connectivity cλ are two important kinds of conditional connectivity, which reflect the number of vertices or edges that can be removed before the graph is disconnected and at least two components contain a cycle, respectively. They have important applications in various networks such as computer networks or biochemical networks. In addition, a fullerene is a special kind of molecule in chemistry. A classic fullerene graph is a 3-connected cubic planar graph with only pentagonal and hexagonal faces. (4, 5, 6)-fullerene graphs are atypical fullerene graphs which also contain 4-faces. In this paper, we prove that cκ=cλ for (4,5,6)-fullerene graphs except for four exceptional graphs with order less than 16. We also give O(ν)-algorithms to determine the cyclic vertex connectivity and the cyclic edge connectivity of (4,5,6)-fullerene graphs.

On the Super (Edge)-Connectivity of Generalized Johnson Graphs

Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.

A Note on Connectivity of Regular Graphs

In this note, we prove the equivalence of edge connectivity, essential edge connectivity and cyclic edge connectivity in an [Formula: see text]-regular graph, where [Formula: see text].

The anti-forcing spectra of [formula omitted]-fullerenes

For an edge subset S of connected graph G, if G−S has only one perfect matching M, then S is called an anti-forcing set of M. The number of edges in a smallest anti-forcing set of M is the anti-forcing number of M, generally indicated by the symbol af(G,M). For a graph G, its anti-forcing spectrum is defined as the integer set Specaf(G):={af(G,M):M is a perfect matching of G}. In this paper, we show that for a (4,6)-fullerene graph Tn with cyclic edge-connectivity 3, Specaf(Tn)=[n+3,2n+4]. Moreover, we show that for any perfect matching M of a (4,6)-fullerene graph G, a minimum anti-forcing set S of M and each M-alternating facial boundary share exactly one edge. Applying this conclusion, we prove that the minimum anti-forcing number of a lantern structure (4,6)-fullerene of order n is ⌊n8⌋+2, and Specaf(Bn)=[⌊n8⌋+2,⌊n3⌋+2].

Super Vertex (Edge)-Connectivity of Varietal Hypercube

The reliability measure of networks is of significant importance to the design and maintenance of networks. Based on connectivity, many refined quantitative indicators for the reliability of network systems have been introduced. The super vertex edge-connectivity and cyclic edge-connectivity, as important parameters to evaluate the robustness of networks, are explored extensively. As a variant of the hypercube Qn, the varietal hypercube VQn has better properties than Qn with the same number of edges and vertices. Wang and Xu have proved that VQn is super vertex-connected for n≥1 and is also super edge-connected if n≠2. In this paper, we use another method to prove these results. Moreover, we also obtain the super restricted connectivity and the cyclic edge-connectivity of the varietal hypercube VQn.

Hamiltonian cycles and 1-factors in 5-regular graphs

It is proven that for any integer g≥0 and k∈{0,…,10}, there exist infinitely many 5-regular graphs of genus g containing a 1-factorisation with exactly k pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For g=0 and k=10, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's “marriage” operation, we discuss two gluing techniques aimed at producing graphs of high cyclic edge-connectivity. We prove that there exist infinitely many planar 5-connected 5-regular graphs in which every 1-factorisation has zero perfect pairs. On the other hand, by the Four Colour Theorem and a result of Brinkmann and the first author, every planar 4-connected 5-regular graph satisfying a condition on its hamiltonian cycles has a linear number of 1-factorisations each containing at least one perfect pair. We also prove that every planar 5-connected 5-regular graph satisfying a stronger condition contains a 1-factorisation with at most nine perfect pairs, whence, every such graph admitting a 1-factorisation with ten perfect pairs has at least two edge-Kempe equivalence classes.

Cyclic connectivity of the data center network

Let G be a connected graph, F be a subset of , S be a subset of . The cyclic vertex connectivity of G, denoted by , is the minimum cardinality of F such that G−F is disconnected and at least two of its components contain cycles. The cyclic edge connectivity of G, denoted by , is the minimum cardinality of S such that G−S is disconnected and at least two of its components contain cycles. Let denote the data center network. In this paper, we obtain the following results: for ; for , ; for ; for ; for , .

Spectral Threshold for Extremal Cyclic Edge-Connectivity

The universal cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth g at least 4, the universal cyclic edge-connectivity is bounded above by \((\Delta -2)g\) where \(\Delta \) is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a d-regular graph of girth \(g\ge 4\) is sufficiently small, then the universal cyclic edge-connectivity is \((d-2)g\), providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight.

Berge–Fulkerson coloring for some families of superposition snarks

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. This conjecture has been verified for many families of snarks with small (≤5) cyclic edge-connectivity. An infinite family, denoted by SK, of cyclically 6-edge-connected superposition snarks was constructed in [European J. Combin. 2002] by Kochol. In this paper, the Berge–Fulkerson conjecture is verified for the family SK, and, furthermore, some larger families containing SK. This is the first paper about the Berge–Fulkerson conjecture for superposition snarks and cyclically 6-edge-connected snarks. Tutte’s integer flow and Catlin’s contractible configuration are applied here as the key methods.

The cubic graphs with finite cyclic vertex connectivity larger than girth

Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity (cλ) has been studied for many years, the study at cyclic vertex connectivity (cκ) is still at the initial stage. And cκ seems to be more complicated than cλ. We have got a sufficient condition that ν(G)≥2g(k−1) for cκ≠∞. On the other hand, if ν(G)<2g(k−1), then we have cκ=∞, or cκ≤(k−2)g, or (k−2)g<cκ<∞. So characterizing all the k-regular graphs with (k−2)g<cκ<∞ is helpful to design an efficient algorithm for cκ. Hence, we characterize all 38 cubic graphs with g<cκ<∞ and prove that cκ=g+1.

Some Results About the Reliability of Folded Hypercubes

In this paper we investigate some reliability measures, including super-connectivity, cyclic edge connectivity, etc., in the folded hypercubes.

Highly-connected planar cubic graphs with few or many Hamilton cycles

In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen (2012) about extremal graphs in these families. In particular, we find families of cyclically 5-edge-connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs P(2n,2). The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes—more precisely, the family consists of the zigzag nanotubes of (fixed) width 5and increasing length. In order to count the Hamilton cycles in the nanotubes, we develop methods inspired by the transfer matrices of statistical physics. We outline how these methods can be adapted to count the Hamilton cycles in nanotubes of greater (but still fixed) width, with the caveat that the resulting expressions involve matrix powers. We also consider cyclically 4-edge-connected planar cubic graphs with few Hamilton cycles, and exhibit an infinite family of such graphs each with exactly 4 Hamilton cycles. Finally we consider the “other extreme” for these two classes of graphs, thus investigating cyclically 4-edge-connected planar cubic graphs with many Hamilton cycles and the cyclically 5-edge-connected planar cubic graphs with few Hamilton cycles. In each of these cases, we present partial results, examples and conjectures regarding the graphs with few or many Hamilton cycles.

The Cyclic Edge-Connectivity of Strongly Regular Graphs

Let G be a connected graph. An edge cut set M of G is a cyclic edge cut set if there are at least two components of $$G-M$$ which contain a cycle. The cyclic edge-connectivity of G is the minimum cardinality of a cyclic edge cut set (if exists) of G. In this paper, we show that the cyclic edge-connectivity of a connected strongly regular graph G (not $$K_{3,3}$$ ) of degree $$k\ge 3$$ with girth c is equal to $$(k-2)c$$ , where $$c=3, 4$$ or 5. Moreover, if G is not the triangular graph srg-(10, 6, 3, 4), the complete multi-partite graph $$K_{2,2,2,2}$$ or the lattice graph srg-(16, 6, 2, 2), then each cyclic edge cut set of size $$(k-2)c$$ is precisely the set of edges incident with a cycle of length c in G.

Flows in Signed Graphs with Two Negative Edges

The presented paper studies the flow number $F(G,\sigma)$ of flow-admissible signed graphs $(G,\sigma)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma)$ there is a set of cubic graphs obtained from $(G,\sigma)$ such that the flow number of $(G,\sigma)$ does not exceed the flow number of any of the cubic graphs. We prove that $F(G,\sigma) \leq 6$ if $(G,\sigma)$ contains a bridge, and $F(G,\sigma) \leq 7$ in general. We prove better bounds, if there is a cubic graph $(H,\sigma_H)$ obtained from $(G,\sigma)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,\sigma) \leq 4$ and the bound is tight. If $H$ is $3$-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,\sigma) \leq 6$. Furthermore, if Tutte's $5$-Flow Conjecture is true, then $(G,\sigma)$ admits a nowhere-zero $6$-flow endowed with some strong properties.

The Cyclic Edge Connectivity and Anti-Kekulé Number of the (5,6,7)-Fullerene

ABSTRACTA plane cubic graph is called a (5,6,7)-fullerene if its faces are only composed of pentagons, hexagons and heptagons. In this paper, we completely characterize the cyclic edge-connectivity of the (5,6,7)-fullerene. Furthermore, we obtain that the anti-Kekulé number of the (5,6,7)-fullerene is 4 when the cyclic edge-connectivity is larger then three. In particular, we obtain some properties with respect to the anti-Kekulé number of the (5,6,7)-fullerene with cyclic 3 edge-connectivity if it is 2-extendable.

Algorithms for Determining the Cyclic Edge Connectivity of Cubic Graphs

In this paper, by proof we directly got the value of cyclic edge connectivity of some cubic graphs such that an algorithm determining the cyclic edge connectivity of cubic graphs was improved. We also added randomness to the algorithm determining cyclic edge connectivity of cubic graphs, although the effect was not ideal. But this method can be used to determine cyclic edge connectivity of general graphs if we can determine the upper bound of cyclic edge connectivity in general graphs. Combining randomized algorithms with the cyclic edge connectivity of graphs is a new attempt.

On Cyclic Edge-Connectivity and Super-Cyclic Edge-Connectivity of Double-Orbit 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 \(\lambda _c(G)\) is the cardinality of a minimum cyclic edge-cut of G. Let \(\zeta (G)=\min \{\omega (X) | X \text{ induce } \text{ a } \text{ shortest } \text{ cycle } \text{ in } G\}\), where \(\omega (X)\) is the number of edges with one end in X and the other end in \(V(G)-X\). A cyclically separable graph G with \(\lambda _c(G)=\zeta (G)\) is said to be cyclically optimal. In particular, we call a graph G super cyclically edge-connected if every minimum cyclic edge-cut isolates a shortest cycle of G. In this work, we first discuss the cyclic edge-connectivity of vertex transitive graphs, regular double-orbit graphs, and the double-orbit graphs with two orbits of same size; moreover, we also discuss the super-cyclic edge-connectivity of double-orbit graphs mentioned above.

Cubic Graphs with Large Circumference Deficit

The circumference cG of a graph G is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4-, 5-, and 6-edge-connected cubic graphs with circumference ratio cG/|VG| bounded from above by 0.876, 0.960, and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 4-edge-connected cubic graph is at least 0.75. Up to our knowledge, no upper bounds on this ratio have been known before for cubic graphs with cyclic edge-connectivity above 3. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hagglund and K. Markstrom, On stable cycles and cycle double covers of graphs with large circumference, Disc Math 312 2012, 2540-2544].

Connectivity Analysis of Cyclically Balanced Fuzzy Graphs

The concepts of connectivity and cycle connectivity play an important role in fuzzy graph theory. In this article, cyclic cutvertices, cyclic bridges and cyclically balanced fuzzy graphs are discussed. It is proved that a vertex in a fuzzy graph is a cyclic cutvertex if and only if it is a common vertex of all strong cycles with maximum strength. Also a cyclic cutvertex cannot be a fuzzy endvertex in a fuzzy graph. A characterization of cyclically balanced fuzzy graphs is obtained. Cyclic vertex connectivity and cyclic edge connectivity of fuzzy graphs are also discussed.