Rong-Xia Hao

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.

Two node-disjoint paths in balanced hypercubes

The balanced hypercube BHn proposed by Wu and Huang is a variation of the hypercube. It has been proved that the balanced hypercube is a node-transitive and bipartite graph. Assume that the nodes are divided into two bipartite node sets X and Y,u and x are two different nodes in X, and v and y are two different nodes in Y. In this paper, we prove that there exist two node-disjoint paths P[x,y] and R[u,v] in BHn, and V(P[x,y])∪V(R[u,v])=V(BHn), where n⩾1. The Hamiltonian laceability of BHn which was proved by Xu et al. is also obtained from the corollary of our result.

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.

Various cycles embedding in faulty balanced hypercubes

Consider the balanced hypercube BH n with | F e | ? 2 n - 3 faulty edges.Prove that every edge of BH n - F e lies on fault-free cycles of even lengths from 6 to 22n.Prove that the lower limit of the length 6 is sharp. Quite a lot of interconnection networks are served as the underlying topologies of large-scale multiprocessor systems. The hypercube is one of the most popular interconnection networks. In this paper we consider the balanced hypercube, which is a variant of the hypercube. Huang and Wu showed that the balanced hypercube has better properties than hypercube with the same number of links and processors. Let F e be the set of faulty edges in an n-dimensional balanced hypercube BH n , where n ? 2 . In this paper, we consider BH n with | F e | ≤ 2 n - 3 faulty edges and prove that every fault-free edge lies on a fault-free cycle of every even length from 6 to 2 2 n in BH n - F e . Furthermore, we prove that the lower limit of the length 6 is sharp by giving a counter example.

Embedding even cycles on folded hypercubes with conditional faulty edges

Let FF e be the set of | FF e | ? 2 n - 4 faulty edges in an n-dimensional folded hypercube FQ n such that each vertex in FQ n is incident to at least two fault-free edges. Under this assumption, we show that every edge of FQ n - FF e lies on a fault-free cycle of every even length from 6 to 2 n , where n ? 5 . Conditional fault means each node is incident to at least two fault-free edges.Consider the folded hypercube FQ n with | FF e | faulty edges under the conditional fault.Prove that every edge of FQ n - FF e lies on cycles of even lengths from 6 to 2 n when | FF e | ? 2 n - 4 , where n ? 5 .

The pessimistic diagnosabilities of some general regular graphs

The pessimistic diagnosis strategy is a classic strategy based on the PMC model. A system is t / t -diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by t p ( G ) , is the maximal number of faulty processors so that the system G is t / t -diagnosable. In this paper, we study the pessimistic diagnosabilities of some general k-regular k-connected graphs G n . The main result t p ( G n ) = 2 k - 2 - g under some conditions is obtained, where g is the maximum number of common neighbors between any two adjacent vertices in G n . As applications of the main result, every pessimistic diagnosability of many famous networks including some known results, such as the alternating group networks AN n , the k-ary n-cubes Q n k , the star graphs S n , the matching composition networks G ( G 1 , G 2 ; M ) and the alternating group graphs AG n , are obtained. We study the pessimistic diagnosabilities of some general regular graphs under the PMC model.The exact value of t p ( G n ) is obtained.As applications, the pessimistic diagnosability of AN n , Q n k , S n , some MCN and AG n etc., are obtained.

Relationship between conditional diagnosability and 2-extra connectivity of symmetric graphs

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability t c ( G ) of G is the maximum number t for which G is conditionally t-diagnosable under the comparison model, while the 2-extra connectivity ? 2 ( G ) of a graph G is the minimum number k for which there is a vertex-cut F with | F | = k such that every component of G - F has at least 3 vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answers this problem by proving t c ( G ) = ? 2 ( G ) for a regular graph G with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, ( n , k ) -star graphs, alternating group networks, ( n , k ) -arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, k-ary n-cube networks, dual-cubes, pancake graphs and hierarchical hypercubes as well. Furthermore, many known results about these networks are obtained directly. We reveal the relationship between conditional diagnosability and 2-extra connectivity of Graphs.The conditional diagnosability under the comparison model is equal to the 2-extra connectivity.As applications, these parameters are determined for some well-known classes of graphs.

The genus distributions of directed antiladders in orientable surfaces

Although there are some results concerning genus distributions of graphs, little is known about those of digraphs. In this work, the genus distributions of 4-regular directed antiladders in orientable surfaces are obtained.

The g-good-neighbor diagnosability of (n,k)-star graphs

Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The g-good-neighbor (conditional) diagnosability such that every fault-free node has at least g fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the PMC model ( 2 ź k ź n - 1 and 1 ź g ź n - k ) and the comparison model ( 2 ź k ź n - 1 and 2 ź g ź n - k ) is n + g ( k - 1 ) - 1 , respectively. In addition, we derive that 1-good-neighbor diagnosability of S n , k under the comparison model is n + k - 2 for 3 ź k ź n - 1 and n ź 4 . As a supplement, we also derive that the g-good-neighbor diagnosability of the ( n , 1 ) -star graph S n , 1 ( 1 ź g ź ź n / 2 ź - 1 and n ź 4 ) under the PMC model and the comparison model is ź n / 2 ź - 1 , respectively. We explore fault diagnosability of multiprocessor systems based on combinatorial network theory.We establish the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the PMC model.We establish the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the comparison model.

The pessimistic diagnosability of three kinds of graphs

A system is t / t -diagnosable if, provided the number of faulty processors is bounded by t , all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G , denoted by t p ( G ) , is the maximal number of faulty processors so that the system G is t / t -diagnosable. The pessimistic diagnosability of alternating group graphs A G n (Tsai, 2015); BC networks (Fan, 2005; Tsai, 2013); the k -ary n -cube networks Q n k , (Wang etźal., 2012); regular graphs including the alternating group networks A N n (Hao etźal., 2016) etc. But most of these results are about networks G with c n ( G ) ź 2 (where c n ( G ) is the maximum number of common neighbors for any two distinct vertices). In this paper, we study the pessimistic diagnosability of three kinds of graphs which are ( n , k ) -arrangement graphs A n , k , ( n , k ) -star graphs S n , k and balanced hypercubes B H n , where c n ( A n , k ) = c n ( S n , k ) = n - k - 1 and c n ( B H n ) = 2 n . We proved that t p ( A n , k ) = ( 2 k - 1 ) ( n - k ) - 1 for n ź k + 2 and k ź 3 , t p ( S n , k ) = n + k - 3 for n ź k + 2 and k ź 3 , and t p ( B H n ) = 2 n for n ź 2 . As corollaries, the pessimistic diagnosability of the known results about A G n and A N n is obtained directly.