Jin-Xin Zhou

Tetravalent half-arc-transitive graphs of order p4

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. It is known that for a prime p there is no tetravalent half-arc-transitive graphs of order p or p^2. Xu [M.Y. Xu, Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified the tetravalent half-arc-transitive graphs of order p^3. As a continuation, we classify in this paper the tetravalent half-arc-transitive graphs of order p^4. It shows that there are exactly p-1 nonisomorphic connected tetravalent half-arc-transitive graphs of order p^4 for each odd prime p.

Conditional Diagnosability of Alternating Group Graphs

Let An be the alternating group of degree n with n ≥ 3. Set S = {(1 2 i), (1 i 2)| 3 ≤ i ≤ n}. The alternating group graph, denoted by AGn, is defined as the Cayley graph on An with respect to S. Jwo et al. [Networks 23 (1993) 315-326] introduced alternating group graph AGn as an interconnection network topology for computing systems. Conditional diagnosability, a new measure of diagnosability introduced by Lai et al. [IEEE Transactions on Computers 54(2) (2005) 165-175] can better measure the diagnosability of regular interconnection networks. This paper determines that under PMC-model the conditional diagnosability of AGn is 4 for n = 4 and 6n -18 for each n ≥ 5.

Hamiltonian cycle embedding for fault tolerance in balanced hypercubes

Abstract The balanced hypercube BH n , defined by Wu and Huang, is a variant of the hypercube network. Yang proposed that fault tolerance of balanced hypercube BH n is an important issue in parallel computing which needs further study (Yang, 2010) [24]. In this paper, we prove that there exists a fault-free Hamiltonian path between any two adjacent vertices in BH n with 2 n - 2 faulty edges. As a corollary, we derive that for any fault-free edge e, there exists a fault-free Hamiltonian cycle containing e in BH n with 2 n - 2 faulty edges which is optimal in the sense of the number of faulty edges.

Tetravalent s-transitive graphs of order 4p

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s+1)-arcs, and 12-arc-transitive if its automorphism group is transitive on vertices, edges but not on arcs. Let p be a prime. Feng et al. [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent half-trasnitive graphs of order 4p, European J. Combin. 28 (2007) 726-733] classified tetravalent 12-arc-transitive graphs of order 4p. In this article a complete classification of tetravalent s-transitive graphs of order 4p is given. It follows from this classification that with the exception of two graphs of orders 8 or 28, all such graphs are 1-transitive. As a result, all tetravalent vertex- and edge-transitive graphs of order 4p are known.

Cubic bi-Cayley graphs over abelian groups

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, some basic properties and the automorphisms of bi-Cayley graphs are explored. As an application, a classification of connected cubic vertex-transitive bi-Cayley graphs over abelian groups is given, and using this, a problem posed in Zhou and Feng (2012) regarding the Cayley property of a class of graphs is solved.

Automorphisms of cubic Cayley graphs of order 2pq

In this paper the automorphism groups of connected cubic Cayley graphs of order 2pq for distinct odd primes p and q are determined. As an application, all connected cubic non-symmetric Cayley graphs of order 2pq are classified and this, together with classifications of connected cubic symmetric graphs and vertex-transitive non-Cayley graphs of order 2pq given by the last two authors, completes a classification of connected cubic vertex-transitive graphs of order 2pq.

Symmetric cubic graphs with solvable automorphism groups

A cubic graph Γ is G-arc-transitive if G≤Aut(Γ) acts transitively on the set of arcs of Γ, and G-basic if Γ is G-arc-transitive and G has no non-trivial normal subgroup with more than two orbits. Let G be a solvable group. In this paper, we first classify all connected G-basic cubic graphs and determine the group structure for every G. Then, combining covering techniques, we prove that a connected cubic G-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type 22, that is, a 2-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order 6.

Semisymmetric elementary abelian covers of the Heawood graph

A regular covering projection @?:X@?->X of connected graphs is said to be elementary abelian if the group of covering transformations is elementary abelian, and semisymmetric if the fibre-preserving group acts edge- but not vertex-transitively. Malnic et al. [A. Malnic, D. Marusic, P. Potocnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 71-97] determined all pairwise nonisomorphic semisymmetric elementary abelian covering projections of the Heawood graph. However, the semisymmetry of covering graphs arising from these covering projections has not been yet verified, which was also pointed out by Conder et al. in [M.D.E. Conder, A. Malnic, P. Potocnik, A census of cubic semisymmetric graphs on up to 768 vertices, J. Algebraic Combin. 23 (2006) 255-294]. In this paper, it is shown that all these covering graphs are indeed semisymmetric, namely, their full automorphism groups are edge- but not vertex-transitive.

Fault-tolerant edge-bipancyclicity of faulty hypercubes under the conditional-fault model

It is well-known that the n-dimensional hypercube Qn is one of the most versatile and efficient interconnection network architecture yet discovered for building massively parallel or distributed systems. Let F be the faulty set of Qn and let fv, fe be the numbers of faulty vertices and faulty edges in F, respectively. An edge e = ( x , y ) is said to be free if e, x, y are not in F, and a cycle is said to be fault-free if there is no faulty vertex or faulty edge on the cycle. In this paper, we prove that each free edge (x, y) in Qn for n ? 3 lies on a fault-free cycle of any even length from 6 to 2 n - 2 f v if f v + f e ? 2 n - 5 , f e ? n - 2 and both x and y are incident to at least two free edges. This result confirms a conjecture reported in the literature.

Super-cyclically edge-connected regular 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. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In Z. Zhang, B. Wang (Super cyclically edge-connected transitive graphs, J. Combin. Optim. 22:549–562, 2011), it is proved that a connected edge-transitive graph is super-λc if either G is cubic with girth at least 7 or G has minimum degree at least 4 and girth at least 6, and the authors also conjectured that a connected graph which is both vertex-transitive and edge-transitive is always super cyclically edge-connected.In this article, for a λc-optimal but not super-λc graph G, all possible λc-superatoms of G which have non-empty intersection with other λc-superatoms are determined. This is then used to give a complete classification of non-super-λc edge-transitive k(k≥3)-regular graphs.