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)

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 ,3q  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 .

(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 ppAVC  qqAVC  atom. by Lemma 2.3, we have that VG is a disjoint 11 2 2  union of distinct  -atoms. c ppVC12  qqVC (2) Let uv,  A, then there exits an automorphism 11 1 AVC  AVC  of G with  u v and so  AA   . By 11 2 2 11   Lemma 2.3,  A  A . Thus the restriction of  on A ppVC12  qqVC induces an automorphism of Y, and then Aut(Y) acts A VC  p  p VC12  q q V C transitively on A1 . Similarly, Aut(Y) acts transitively on  A2 . X1 and X 2 are two orbits of Aut(G). By (1), there GAC p pVC 12   qqVC exists  i Aut G i 1, , m, such that m  G C . VG  i  A. i1 a contradiction. Theorem 3.2 Let G be a connected pq, -half vertex Since Aut(G) has two orbits X1 and X 2 , for any transitive graph with bipartition X12 X ,  G  4 1,ij m and ij , ij A1   A1  and and girth g  6 , then G is super- c . m Proof. By Theorem 3.1, G is c -optimal. Suppose G A , AX . Thus, we have X   A , ij11  1 1 i 1is not super- c . By Lemma 2.8, G has a super atom. By i1 Lemma 2.9, every super atom is impimitive block. Let A m be a super atom of G, by Lemma 2.3, GA  is half- X 22  i A , and XmAiii1, 2 . Thus Y is a i1 vertex transitive. Let A11 AX and A22 AX ,

pq, -half vertex transitive graph, where then A  AA12 . Suppose GA  is  pq,  by 31,3pp qq1 (by Lemma 2.4). Lemma 2.4 (2), pq,3  . Let C be a shorte st cyc le of Lemma 2.8 ([17]) A cyclically optimal graph is not GA  . With a similar proof as Th eorem 3.1, we can get super cyclically edge-connected if and only if it has a  A  G C , a contradiction. super atom. Lemma 2.9 Let G be a connected (p,q)-half vertex 4. Acknowledgements transitive graph with bipartition XX12 ,4  G  and girth g G . Suppose A is a super atom of G and We would like to appreciate the anonymous referees for the valuable suggestions which help us a lot in refining YGA   . If G is c -optimal but not super- c , then (1) VG is a disjoint union of distinct super atoms; the presentation of this paper. (2) Y is a  pq,  -half vertex transitive graph, where 31pp,3 q  q1 REFERENCES With a similar argument as the proof of Lemma 2.7, [1] S. Latifi, M. Hegde and M. Naraghi-Pour, “Conditional we can prove it. Connectivity Measures for Large Multiprocessor Sys- tems,” IEEE Transactions on Compututers, Vol. 43, No. 3. Super-λc Half Vertex Transitive Graphs 2, 1994, pp. 218-222. doi:10.1109/12.262126 Theorem 3.1 Let G be a connected (p,q)-half vertex [2] L. Lovász, “On Graphs Not Containing Independent Cir- cuits,” Matematikai Lapok, Vol. 16, No. 3, 1965, pp. 289- transitive graph with bipartition X  X ,  G  4 12   299. and girth g  6 , then G is c -optimal. Proof. By Lemma 2.1, G is cyclically separable. [3] B. Bollobás, “Extremal Graph Theory,” Academic Press, London, 1978. Suppose G is not c -optimal. By Lemma 2.2, every atom is impimitive block. Let A be a atom of G, by [4] M. D. Plummer, “On the Cyclic Connectivity of Planar Graphs,” Lecture Notes in Mathematics, Vol. 303, No. 1, Lemma 2.3, GA is half-vertex transitive. Let   1972, pp. 235-242. doi:10.1007/BFb0067376 A11 AX and A22 AX , then A  AA12 . Suppose GA is pq, by Lemma 2.4 (2), [5] P. G. Tait, “Remarks on the Colouring of Maps,” Pro-     ceedings of the Royal Society of Edinburgh, Vol. 10, No. pq, 3. Let C be a shortest cycle of GA  . Then by 4, 1880, pp. 501-503. Lemma 2.4 (2) and Lemma 2.5, GA contains two   [6] E. Máčajová and M. Šoviera, “Infinitely Many Hypoham- disjoint cycles, and VC, AVC is s cyclic edge- iltonian Cubic Graphs of Girth 7,” Graphs and Combina- torics, Vol. 27, No. 2, 2011, pp. 231-241. cut. Clearly, A VC A  C since no two vertices doi:10.1007/s00373-010-0968-z of C have common neighbor in A VC. Then, [7] F. Kardoš and R. Šrekovski, “Cyclic Edge-Cuts in Fuller-

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