VERTEX-FAULT-TOLERANT CYCLES EMBEDDING ON ENHANCED HYPERCUBE NETWORKS

In this paper, we study the enhanced hypercube, an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube, and focus on cycles embedding on the enhanced hypercube with faulty vertices. Let Fu be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k （n ≥ 3, 1 ≤ k 〈≤n - 1）. When IFvl = 2, we showed that Qn,k - Fv contains a fault-free cycle of every even length from 4 to 2n - 4 where n （n ≥ 3） and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2n - 4, simultaneously, contains a cycle of every odd length from n-k ＋ 2 to 2^n-3 where n （≥ 3） and k have the different parity. Furthermore, when ｜Fv｜ = fv ≤ n - 2, we prove that there exists the longest fault-free cycle, which is of even length 2^n - 2fv whether n （n ≥ 3） and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2^n - 2fv ＋ 1 in Qn,k - Fv where n （≥ 3） and k have the different parity.     

CYCLES EMBEDDING ON FOLDED HYPERCUBES WITH FAULTY NODES

Let FF_v be the set of faulty nodes in an n-dimensional folded hypercube FQ_n with |FF_v| ≤ n-1 and all faulty vertices are not adjacent to the same vertex. In this paper, we show that if n ≥ 4, then every edge of FQn-FF_v lies on a fault-free cycle of every even length from 6 to 2~n-2|FF_v|.     

ON THE CONDITIONAL EDGE CONNECTIVITY OF ENHANCED HYPERCUBE NETWORKS

Let G =(V, E) be a connected graph and m be a positive integer, the conditional edge connectivity λ_δ~m is the minimum cardinality of a set of edges,if it exists, whose deletion disconnects G and leaves each remaining component with minimum degree δ no less than m. This study shows that λ_δ~1(Q_(n,k)) = 2 n,λ_δ~2(Q_(n,k)) = 4 n-4(2 ≤ k ≤ n-1, n ≥ 3) for n-dimensional enhanced hypercube Q_(n,k). Meanwhile, another easy proof about λ_δ~2(Q_n) = 4 n-8, for n ≥ 3 is proposed. The results of enhanced hypercube include the cases of folded hypercube.     

Transitivity of varietal hypercube networks

The varietal hypercube VQn is a variant of the hypercube Qn and has better properties than Qn with the same number of edges and vertices. This paper proves that VQn is vertex-transitive. This property shows that when VQn is used to model an interconnection network, it is high symmetrical and obviously superior to other variants of the hypercube such as the crossed cube.     

中心对称本原矩阵的k点指数集

Let G = (V, E) be a primitive digraph. The vertex exponent of G at a vertex v ∈ V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. We choose to order the vertices of G in the k-point exponent of G and is denoted by expG(k), 1 ≤ k ≤ n. We define the k-point exponent set E(n, k) := {expG(k)| G = G(A) with A ∈ CSP(n)}, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n,k) for all n, k with 1 ≤ k ≤ n except n ≡ 1(mod 2) and 1 ≤ k ≤ n - 4. We also characterize the extremal graphs when k = 1.     

Neighbor Sum Distinguishing Colorings of Graphs with Maximum Average Degree Less Than 37/12

Let G be a graph and let its maximum degree and maximum average degree be denoted byΔ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ'∑≠(G). Flandrin et al. proposed the following conjecture thatχ'∑(G) ≤Δ(G) + 2 for any connected graph with at least 3 vertices and G≠C_5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) 37/12 and Δ(G) ≥ 7.     

The k-point Exponent Set of Primitive Digraphs with Girth 2

Let D =(V,E)be a primitive digraph.The vertex exponent of D at a vertex v∈V,denoted by exPD(V),is the least integer p such that there is a v→u walk of length p for each u∈V.Following Brualdi and Liu,we order the vertices of D so that exPD(v_1)≤exPD(v_2)≤…≤exPD(v_n).Then exPD(v_k)is called the k- point exponent of D and is denoted by exP_D(k),1≤k≤n.In this paper we define e(n,k):=max{exp_D(k)|D∈PD(n,2)} and E(n,k):= {expD(k)|D∈PD(n,2)},where PD(n,2)is the set of all primitive digraphs of order n with girth 2.We completely determine e(n,k)and E(n,k)for all n,k with n≥3 and 1≤k≤n.     

Restricted Fault Diameter of Hypercube Networks

This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n ≥ 2).It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2n-3 vertices in Qn - {x, y}, if F contains neither of neighbor-sets of x and y in Qn, then the distance between x andy in Qn - F is given by D(Qn-F;x,y){=1 , for=1;≤d 4 , for 2≤d≤n-2,n≥4;≤n 1, for d=n-1,n≥3; =n, for d=n. Furthermore, the upper bounds are tight. As an immediately consequence, Qn can tolerate up to 2n-3 vertices failures and remain diameter 4 if n = 3 and n 2 if n ≥ 4 provided that for each vertex x in Qn, all the neighbors of x do not fail at the same time. This improves Esfahanian‘s result.     

Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs K4,n and Kn,n

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

关于完全二部图$K_{4, n}$和$K_{n, n}$的点可区别 $IE$-全染色的一些注记

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that every edge of Q_n-F_v-F_e lies on a cycle of every even length from 4 to 2ⁿ-2|F_v| even if |F_v|+|F_e|?n-2, where n?3. Since Q_n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.     

Vertex-fault-tolerant cycles embedding on enhanced hypercube networks

In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F _v be the set of faulty vertices in the n-dimensional enhanced hypercube Q _n,k (1 ≤ k ≤ n?1). When |F _v | = 2, we showed that Q _n,k ? F _v contains a fault-free cycle of every even length from 4 to 2 ⁿ ?4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2 ⁿ ? 4, simultaneously, contains a cycle of every odd length from n ? k + 2 to 2 ⁿ ? 3 where n(≥ 3) and k have the different parity. Furthermore, when |F ^v | = f ^v ≤ n ? 2, we proof that there exists the longest fault-free cycle, which is of even length 2 ⁿ ? 2f ^v whether n(n ≥ 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2 ⁿ ? 2f ^v ? 1 in Q ^n,k ? F ^v where n(≥ 3) and k have the different parity.     

1-vertex-fault-tolerant cycles embedding on folded hypercubes

In this paper, we focus on a hypercube-like structure, the folded hypercube, which is basically a standard hypercube with some extra links between its nodes. Let f be a faulty vertex in an n-dimensional folded hypercube FQ_n. We show that FQ_n−{f} contains a fault-free cycle of every even length from 4 to 2ⁿ−2 if n≥3 and, furthermore, every odd length from n+1 to 2ⁿ−1 if n≥2 and n is even.     

含故障点的加强超立方体中路和圈的嵌入

本文研究了含故障点的n-维加强超立方体Q_n,k中的路和圈嵌入的问题.充分分析了加强超立方体网络的潜在特性,利用了构造的方法.得到了含2n-4个故障点的加强超立方体Q_n,k中含长为2ⁿ-2f的容错圈的结论,推广了折叠超立方体网络中1-点容错圈嵌入的结果.其中折叠超立方体网络为加强超立方体网络的一种特殊情况.     

Fault-tolerant panconnectivity of augmented cubes

The augmented cube AQ _n is a variation of the hypercube Q _n . This paper considers the panconnectivity of AQ _n (n ⩾ 3) with at most 2n−5 faulty vertices and/or edges and shows that, for any two fault-free vertices u and v with distance d in AQ _n , there exist fault-free uv-paths of every length from d + 2 to 2 ⁿ − f − 1, where f is the number of faulty vertices in AQ _n . The proof is based on an inductive construction.     

Embedding long cycles in faulty k-ary 2-cubes

The class of k-ary n-cubes represents the most commonly used interconnection topology for distributed-memory parallel systems. Given an even k ? 4, let (V₁, V₂) be the bipartition of the k-ary 2-cube, f_v1, f_v2 be the numbers of faulty vertices in V₁ and V₂, respectively, and f_e be the number of faulty edges. In this paper, we prove that there exists a cycle of length k² − 2max{f_v1, f_v2} in the k-ary 2-cube with 0 ? f_v1 + f_v2 + f_e ? 2. This result is optimal with respect to the number of faults tolerated.     

Cycle embedding in star graphs with more conditional faulty edges

The star graph is one of the most attractive interconnection networks. The cycle embedding problem is widely discussed in many networks, and edge fault tolerance is an important issue for networks since edge failures may occur when a network is put into use. In this paper, we investigate the cycle embedding problem in star graphs with conditional faulty edges. We show that there exist fault-free cycles of all even lengths from 6 to n! in any n-dimensional star graph S_n (n ? 4) with ?3n − 10 faulty edges in which each node is incident with at least two fault-free edges. Our result not only improves the previously best known result where the number of tolerable faulty edges is up to 2n − 7, but also extends the result that there exists a fault-free Hamiltonian cycle under the same condition.     

A note on short cycles in a hypercube

How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd?s about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,C_l) be the largest number of edges in a subgraph of a hypercube Q_n containing no cycle of length l. It is known that f(n,C_l)=o(|E(Q_n)|), when l=4k, k?2 and that . It is an open question to determine f(n,C_l) for l=4k+2, k?2. Here, we give a general upper bound for f(n,C_l) when l=4k+2 and provide a coloring of E(Q_n) by four colors containing no induced monochromatic C₁₀.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).