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Super Cyclically Edge Connected Half Vertex Transitive Graphs* Applied Mathematics, 2013, 4, 348-351 http://dx.doi.org/10.4236/am.2013.42053 Published Online February 2013 (http://www.scirp.org/journal/am) Super Cyclically Edge Connected Half Vertex * Transitive Graphs Haining Jiang, Jixiang Meng#, Yingzhi Tian College of Mathematics and System Sciences, Xinjiang University, Urumqi, China Email: [email protected], #[email protected], [email protected] Received October 21, 2012; revised December 26, 2012; accepted January 3, 2013 ABSTRACT Tian and Meng in [Y. Tian and J. Meng, c -Optimally half vertex transitive graphs with regularity k , Information Processing Letters 109 (2009) 683-686] shown that a connected half vertex transitive graph with regularity k and girth gG 6 is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree G 4 and girth gG 6 . Keywords: Cyclic Edge-Connectivity; Cyclically Optimal; Super Cyclically Edge-Connected; Half Vertex Transitive Graph 1. Introduction cuts of G by following Plummer [4]. The concept of cyclic edge-connectivity as applied to planar graphs dates The traditional connectivity and edge-connectivity, are to the famous incorrect conjecture of Tait [5]. important measures for networks, which can correctly The cyclic edge-connectivity plays an important role reflect the fault tolerance of systems with few processors, in some classic fields of graph theory such as Hamil- but it always underestimates the resilience of large net- tonian graphs (Máčajová and Šoviera [6]), fullerence works. The discrepancy incurred is because events whose graphs (Kardoš and Šrekovski [7]), integer flow con- occurrence would disrupt a large network after a few jectures (Zhang [8]), n-extendable graphs (Holton et al. processors, therefore, the disruption envisaged occurs in [9]; Lou and Holton [10]), etc. a worst case scenario. To overcome such a shortcoming, For two vertex sets X ,,,YVXY is the set of Latifi et al. [1] proposed a kind of conditional edge- G edges with one end in X and the other end in Y . For connectivity, denoted by k G , which is the minimum any vertex set X , GX is the subgraph of G induced size of edge-cut S such that each vertex has degree at by X , X is the complement of X . Clearly, if least k in G S. X , X is a minimum cyclic edge-cut, then both Throughout the paper graphs are undirected finite connected without loops or multiple edges. GX and GX are connected. We set Let GVE , be a graph, an edge set F is a GXX min induces a shortest cycle in G, cyclic edge-cut if GF is disconnected and at least two of its components contain cycles. Clearly, a graph where X is the number of edges with one end in has a cyclic edge-cut if and only if it has two vertex- X and the other end in VG\ X. It has been proved disjoint cycles. A graph G is said to be cyclically se- in Wang and Zhang [11] that c G G for any parable if G has a cyclic edge-cut. Note that Lovász [2] cyclically separable graph. Hence, a cyclically separable characterized all multigraphs without two vertex-disjoint graph G is called cyclically optimal, in short, c -optimal , cycles. The characterization can also be found in Bollo- if c G G , and super cyclically edge-connected, bás [3]. So, it is natural to further study the cyclically se- in short, super-c , if the removal of any minimum cyclic parable graphs. For a cyclically separable graph G , The edge-cut of graph G results in a component which is a cyclic edge-connectivity of G , denoted by c G , is shortest cycle. defined as the minimum cardinality over all cyclic edge- Cyclic edge-fragment and cyclic edge-atom play a fundamental role. A vertex set X is a cyclic edge- *This research is supported by NSFC (10671165) and NSFCXJ (2010211A06). fragment, in short, fragment, if X , X is a minimum #Corresponding author. cyclic edge-cut. A cyclic edge-fragment with the mini- Copyright © 2013 SciRes. AM H. N. JIANG ET AL. 349 mum cardinality is called a cyclic edge-atom, in short, q)-biregular graph with pq 4. Suppose G is not atom. A cyclic edge-fragment of G is said to be super, cyclically optimal and gG 6 . Then for any distinct if neither X nor X induces a shortest cycle, in short, atoms X and Y, XY . super fragment. A super cyclic edge-fragment with the An imprimitive block of G is a proper nonempty minimum cardinality is called a super cyclic edge-atom, subset A of VG such that for any automorphism in short, super atom. A cyclic edge-fragment is said to be Aut G , either A A or A . trivial, if it induces a cycle, otherwise it is nontrivial. Lemma 2.3 ([20]) Let GV , E be a graph and let A graph G is said to be vertex transitive if Aut G Y be the subgraph of G induced by an imprimitive block acts transitively on VG, and is edge transitive if A of G. If G is vertex-transitive, then so is Y. If G is Aut G acts transitively on EG. A bipartite graph is edge-transitive, then A is an independent set of G. biregular, if all the vertices from the same partite set If X is a super atom, and X is a proper subset of X have the same degree. We abbreviate the bipartite graph such that X , X is a cyclic edge-cut and GX is as a pq, -biregular graph, if the two distinct degrees not a shortest cyclic, then are p and q respectively p q . A bipartite graph G with bipartition X1 X 2 is called half vertex tran- X X . sitive [12], if Aut G acts transitively both on X1 and The observation is used frequently in the proofs. X 2 . Clearly, the half vertex transitive graph is biregular graph. Let x VG, we call the set xg : gAutG Lemma 2.4 ([11]) Let G be a connected graph with G 3 and X be a fragment. Then an orbit of Aut G . Clearly, Aut G acts transitively (1) GX 2 ; on each orbit of Aut G . Transitive graphs have been playing an important role in designing network to- (2) If GX 3 , then dvX dvX holds for pologies, since they possess many desirable properties any v X; such as high fault tolerance, small transitive delay, etc. (3) If GX is not a cycle and v is a vertex in X [13,14]. with dvX 2 , then dvX dvX holds for any In Nedela and Škoviera [15], it was proved that a cubic- v X; transitive or edge-transitive graph (expect for K and 4 (4) If G 4 , and X is a non-trivial atom of G, K ) is -optimal. From Wang and Zhang [11], Xu 3,3 c then GX 3. Furthermore, dv dv holds and Liu [16], we have known that a k 4-regular X X for any v X. and dvX dv holds for any vertex-transitive graph G is c -optimal if it has girth X gG 5 . It was also shown that an edge-transitive vX . graph G with minimum degree G 4 and order Lemma 2.5 ([17]) Let G be a connected graph with n 6 is c -optimal in Wang and Zhang [11]. Recently, G 3 and gG 6 . Then G has two vertex-dis- Zhang and Wang [17] showed that a connected vertex- joint cycles and VG G12 g g. transitive or edge-transitive graph is super- c if either G is cubic with girth gG 7 or G has minimum Lemma 2.6 ([17]) Let G be a (p,q)-biregular graph degree G 4 and girth gG 6 . Zhou and Feng with G 4 and girth gG 6 . Suppose G is cyclically optimal but not super cyclically edge-connect- [18] characterized all possible c -superatoms for c - optimal nonsuper- graphs, and classified all -op- ed. Then any two distinct super atoms X and Y of G c c satisfies XY . timal nonsuper- c edge-transitive graphs. Theorem 1.1 ([19]) Let G be a k 4-regular con- Lemma 2.7 Let G be a connected (p,q)-half vertex nect half vertex transitive graph with bipartition X X , transitive graph with bipartition X12X , G 4 and 12 girth g G . Suppose A is a atom of G and YG A . and girth g 6 , then G is c -optimal. Motivated by the work in Tian and Meng [19], in this If G is not c -optimal, then article we aim to study a connected half vertex transitive (1) VG is a disjoint union of distinct atoms; graph, and we show that a connected half vertex tran- (2) Y is a pq, -half vertex transitive graph, where sitive graph G is super cyclically edge-connected if 31 pp ,3q q1. minimum degree G 4 and girth gG 6 . Proof. Let A11 AX and A22 AX , 2. Preliminaries then Lemma 2.1 ([11]) Let G be a simple connected graph A AA12 . with G 3 and gG 5 or G 4 and or- Since A is a c -atom, we have der n 6 . Then G is cyclically separable. Lemma 2.2 ([11]) Let G be a cyclically separable (p, Aii 21 2 . Copyright © 2013 SciRes. AM 350 H. N. JIANG ET AL. (1) Since Ai 1, 2 and Aut(X) acts transitively i AppAqq 12 A both on X and X , each vertex of G lies in a - 1 2 c ppAVC qqAVC atom.
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