ABSTRACT

Tian and Meng in [Y. Tian and J. Meng, -Optimally half vertex transitive graphs with regularity, Information Processing Letters 109 (2009) 683-686] shown that a connected half vertex transitive graph with regularity and girth is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree and girth.

1. Introduction

The traditional connectivity and edge-connectivity, are important measures for networks, which can correctly reflect the fault tolerance of systems with few processors, but it always underestimates the resilience of large networks. The discrepancy incurred is because events whose occurrence would disrupt a large network after a few processors, therefore, the disruption envisaged occurs in a worst case scenario. To overcome such a shortcoming, Latifi et al. [1] proposed a kind of conditional edgeconnectivity, denoted by, which is the minimum size of edge-cut such that each vertex has degree at least in.

Throughout the paper graphs are undirected finite connected without loops or multiple edges.

Let be a graph, an edge set is a cyclic edge-cut if is disconnected and at least two of its components contain cycles. Clearly, a graph has a cyclic edge-cut if and only if it has two vertexdisjoint cycles. A graph is said to be cyclically separable if has a cyclic edge-cut. Note that Lovász [2] characterized all multigraphs without two vertex-disjoint cycles. The characterization can also be found in Bollobás [3]. So, it is natural to further study the cyclically separable graphs. For a cyclically separable graph, The cyclic edge-connectivity of, denoted by, is defined as the minimum cardinality over all cyclic edgecuts of by following Plummer [4]. The concept of cyclic edge-connectivity as applied to planar graphs dates to the famous incorrect conjecture of Tait [5].

The cyclic edge-connectivity plays an important role in some classic fields of graph theory such as Hamiltonian graphs (Máčajová and Šoviera [6]), fullerence graphs (Kardoš and Šrekovski [7]), integer flow conjectures (Zhang [8]), n-extendable graphs (Holton et al. [9]; Lou and Holton [10]), etc.

For two vertex sets is the set of edges with one end in and the other end in. For any vertex set, is the subgraph of induced by, is the complement of. Clearly, if is a minimum cyclic edge-cut, then both

and are connected. We set

where is the number of edges with one end in and the other end in. It has been proved in Wang and Zhang [11] that for any cyclically separable graph. Hence, a cyclically separable graph G is called cyclically optimal, in short, , if, and super cyclically edge-connected, in short, , if the removal of any minimum cyclic edge-cut of graph results in a component which is a shortest cycle.

Cyclic edge-fragment and cyclic edge-atom play a fundamental role. A vertex set is a cyclic edgefragment, in short, fragment, if is a minimum cyclic edge-cut. A cyclic edge-fragment with the minimum cardinality is called a cyclic edge-atom, in short, atom. A cyclic edge-fragment of is said to be super, if neither nor induces a shortest cycle, in short, super fragment. A super cyclic edge-fragment with the minimum cardinality is called a super cyclic edge-atom, in short, super atom. A cyclic edge-fragment is said to be trivial, if it induces a cycle, otherwise it is nontrivial.

A graph is said to be vertex transitive if acts transitively on, and is edge transitive if acts transitively on. A bipartite graph is biregular, if all the vertices from the same partite set have the same degree. We abbreviate the bipartite graph as a -biregular graph, if the two distinct degrees are and respectively. A bipartite graph with bipartition is called half vertex transitive [12], if acts transitively both on and. Clearly, the half vertex transitive graph is biregular graph. Let, we call the set

an orbit of. Clearly, acts transitively on each orbit of. Transitive graphs have been playing an important role in designing network topologies, since they possess many desirable properties such as high fault tolerance, small transitive delay, etc. [13,14].

In Nedela and Škoviera [15], it was proved that a cubictransitive or edge-transitive graph (expect for and) is -optimal. From Wang and Zhang [11], Xu and Liu [16], we have known that a -regular vertex-transitive graph is -optimal if it has girth. It was also shown that an edge-transitive graph with minimum degree and order is -optimal in Wang and Zhang [11]. Recently, Zhang and Wang [17] showed that a connected vertextransitive or edge-transitive graph is super- if either is cubic with girth or G has minimum degree and girth. Zhou and Feng [18] characterized all possible -superatoms for - optimal nonsuper- graphs, and classified all -optimal nonsuper- edge-transitive graphs.

Theorem 1.1 ([19]) Let G be a -regular connect half vertex transitive graph with bipartition, and girth, then G is -optimal.

Motivated by the work in Tian and Meng [19], in this article we aim to study a connected half vertex transitive graph, and we show that a connected half vertex transitive graph is super cyclically edge-connected if minimum degree and girth.

2. Preliminaries

Lemma 2.1 ([11]) Let G be a simple connected graph with and or and order. Then G is cyclically separable.

Lemma 2.2 ([11]) Let G be a cyclically separable (p, q)-biregular graph with. Suppose G is not cyclically optimal and. Then for any distinct atoms X and Y,.

An imprimitive block of is a proper nonempty subset of such that for any automorphism, either or.

Lemma 2.3 ([20]) Let be a graph and let Y be the subgraph of G induced by an imprimitive block A of G. If G is vertex-transitive, then so is Y. If G is edge-transitive, then A is an independent set of G.

If X is a super atom, and is a proper subset of X such that is a cyclic edge-cut and is not a shortest cyclic, then

The observation is used frequently in the proofs.

Lemma 2.4 ([11]) Let G be a connected graph with and be a fragment. Then

(1);

(2) If, then holds for any;

(3) If is not a cycle and is a vertex in X with, then holds for any;

(4) If, and X is a non-trivial atom of Gthen. Furthermore, holds for any. and holds for any.

Lemma 2.5 ([17]) Let G be a connected graph with and. Then G has two vertex-disjoint cycles and.

Lemma 2.6 ([17]) Let G be a (p,q)-biregular graph with and girth. Suppose G is cyclically optimal but not super cyclically edge-connected. Then any two distinct super atoms X and Y of G satisfies.

Lemma 2.7 Let G be a connected (p,q)-half vertex transitive graph with bipartition, and girth. Suppose A is a atom of G and. If G is not -optimal, then

(1) is a disjoint union of distinct atoms;

(2) Y is a -half vertex transitive graph, where.

Proof. Let

andthen

.

Since A is a -atom, we have

.

(1) Since and Aut(X) acts transitively both on and, each vertex of G lies in a - atom. by Lemma 2.3, we have that is a disjoint union of distinct -atoms.

(2) Let, then there exits an automorphism

of G with and so. By Lemma 2.3,. Thus the restriction of on A induces an automorphism of Y, and then Aut(Y) acts transitively on. Similarly, Aut(Y) acts transitively on. and are two orbits of Aut(G). By (1), there exists, such that

Since Aut(G) has two orbits and, for any and, and

. Thus, we have, , and. Thus Y is a

-half vertex transitive graph, where

(by Lemma 2.4).

Lemma 2.8 ([17]) A cyclically optimal graph is not super cyclically edge-connected if and only if it has a super atom.

Lemma 2.9 Let G be a connected (p,q)-half vertex transitive graph with bipartition and girth. Suppose A is a super atom of G and. If G is -optimal but not super-, then

(1) is a disjoint union of distinct super atoms;

(2) Y is a -half vertex transitive graph, where

With a similar argument as the proof of Lemma 2.7, we can prove it.

3. Super-λ_c Half Vertex Transitive Graphs

Theorem 3.1 Let G be a connected (p,q)-half vertex transitive graph with bipartition, and girth, then G is -optimal.

Proof. By Lemma 2.1, G is cyclically separable. Suppose G is not -optimal. By Lemma 2.2, every atom is impimitive block. Let A be a atom of G, by Lemma 2.3, is half-vertex transitive. Let

and, then.

Suppose is by Lemma 2.4 (2),

. Let C be a shortest cycle of. Then by Lemma 2.4 (2) and Lemma 2.5, contains two disjoint cycles, and is s cyclic edgecut. Clearly, since no two vertices of C have common neighbor in. Then,

a contradiction.

Theorem 3.2 Let G be a connected -half vertex transitive graph with bipartition, and girth, then G is super-.

Proof. By Theorem 3.1, G is -optimal. Suppose G is not super-. By Lemma 2.8, G has a super atom. By Lemma 2.9, every super atom is impimitive block. Let A be a super atom of G, by Lemma 2.3, is halfvertex transitive. Let andthen. Suppose is by Lemma 2.4 (2),. Let C be a shortest cycle of. With a similar proof as Theorem 3.1, we can get

, a contradiction.

4. Acknowledgements

We would like to appreciate the anonymous referees for the valuable suggestions which help us a lot in refining the presentation of this paper.

REFERENCES

NOTES

^*This research is supported by NSFC (10671165) and NSFCXJ (2010211A06).

^#Corresponding author.