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.     

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.     

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.     

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.     

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, ...).

     

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.     

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.     

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.     

A Ramsey‐type result for the hypercube

We prove that for every fixed k and ? ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Q_n with k colors contains a monochromatic cycle of length 2 ?. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, that is, have the property that for every k, any k‐edge coloring of a sufficiently large Q_n contains a monochromatic copy of H. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 196–208, 2006     

Kantorovič operators of second order

The Kantorovi? operators of second order are introduced byQ _n f= =(B _n+2 F)″ whereF is the double indefinite integraloff andB _n+2 the (n+2)-th Bernstein operator. The operatorsQ _n will reveal a close affinity to the so-called modified Bernstein operatorsC _n introduced bySchnabl [10] on a quite different way. The article contains investigations concerning the asymptotic behavior ofQ _n ^kn f (asn → ∞), where (k _n) is a sequence of natural numbers.     

Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {?1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k ? 1. That is, Σ _x∈N[v] f(x) ≤ k ? 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σ _v∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α _s ^k (G) of G. In this work, we mainly present upper bounds on α _s ^k (G), as for example α _s ^k (G) ≤ n ? 2?(Δ(G) + 2 ? k)/2?, and we prove the Nordhaus-Gaddum type inequality $\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$ , where n is the order, Δ(G) the maximum degree and $\bar G$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.     

Sums of roots of unity in cyclotomic fields

Let ζ_n denote a primitive nth root of unity, n ≥ 4. For any integer k, 2 ≤ k ≤ n ? 2 it is shown that the diophantine equation 1 + ζ_n + ? + ζ_n^k?1 = ?α^q has no solutions with ?, α in $\text{Q}$(ζ_n), ? a root of unity, α an algebraic integer, and q an integer ≥ 2, except when n = 10, 12, or 30, where the solutions are completely determined.     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

Embedding cycles of various lengths into star graphs with both edge and vertex faults

The n-dimensional star graph S_n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F_v and F_e be the sets of faulty vertices and faulty edges of S_n, respectively. We prove that S_n − F_v − F_e contains a fault-free cycle of every even length from 6 to n! − 2∣F_v∣ with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4. We also show that S_n − F_v − F_e contains a fault-free path of length n! − 2∣F_v∣ − 1 (respectively, n! − 2∣F_v∣ − 2) between two arbitrary vertices of S_n in different partite sets (respectively, the same partite set) with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4.     

Shortest string containing all permutations

In this paper, we consider the problem of constructing a shortest string of {1,2,…,n} containing all permutations on n elements as subsequences. For example, the string 1 2 1 3 1 2 1 contains the 6 (=3!) permutations of {1,2,3} and no string with less than 7 digits contains all the six permutations. Note that a given permutation, such as 1 2 3, does not have to be consecutive but must be from left to right in the string.We shall first give a rule for constructing a string of {1,2,…,n} of infinite length and the show that the leftmost n²?2n+4 digits of the string contain all the n! permutations (for n≥3). We conjecture that the number of digits f(n) = n²?2n+4 (for n≥3) is the minimum.Then we study a new function F(n,k) which is the minimum number of digits required for a string of n digits to contain all permutations of i digits, i≤k. We conjecture that F(n,k) = k(n?1) for 4≤k ≤n?1.     

Cycles embedding on folded hypercubes with faulty nodes

Let F_Fv$F{F}_v$ be the set of faulty nodes in an n$n$-dimensional folded hypercube F_Qn$F{Q}_n$ with |F_Fv|≤n−2$\left|F{F}_v\right|\le n-2$. In this paper, we show that if n≥3$n\ge 3$, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every even length from 4$4$ to 2ⁿ−2|F_Fv|$2^n-2\left|F{F}_v\right|$, and if n≥2$n\ge 2$ and n$n$ is even, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every odd length from n+1$n+1$ to 2ⁿ−2|F_Fv|−1$2^n-2\left|F{F}_v\right|-1$.     

Decomposition of the completer-graph into completer-partiter-graphs

Forn ≥ r ≥ 1, letf _r (n) denote the minimum numberq, such that it is possible to partition all edges of the completer-graph onn vertices intoq completer-partiter-graphs. Graham and Pollak showed thatf ₂(n) =n ? 1. Here we observe thatf ₃(n) =n ? 2 and show that for every fixedr ≥ 2, there are positive constantsc ₁(r) andc ₂(r) such thatc ₁(r) ≤f _r (n)?n ^?[r/2] ≤n ₂(r) for alln ≥ r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C _k denote a cycle of length k, and let C _k denote the set of cycles C _?, where 3≤?≤k and ? and k have the same parity. Erd?s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C _k ) ～ z(n,F) — here we write f(n) ～ g(n) for functions f,g: ? → ? if lim _n→∞ f(n)/g(n)=1. They proved this when F ={C ₄} by showing that ex(n,{C ₄;C ₅})～z(n,C ₄). In this paper, we extend this result by showing that if ?∈{2,3,5} and k>2? is odd, then ex(n,C _2? ∪{C _k }) ～ z(n,C _2? ). Furthermore, if k>2?+2 is odd, then for infinitely many n we show that the extremal C _2? ∪{C _k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2?, and furthermore the asymptotic result does not hold when (?,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.