Uncountable Homogeneous Partial Orders

A partially ordered set (P, ≤) is called k‐homogeneous if any isomorphism between k‐element subsets extends to an automorphism of (P, ≤). Assuming the set‐theoretic assumption ⋄(ϰ₁), it is shown that for each k, there exist partially ordered sets of size ϰ₁ which embed each countable partial order and are k‐homogeneous, but not (k + 1)‐homogeneous. This is impossible in the countable case for k ≥ 4.     

More results on perfect (3,3,6k + 4; 6k − 2)‐threshold schemes

A (2,3)‐packing on X is a pair (X, ), where is a set of 3‐subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X,E) such that E consists of all the pairs which do not appear in any block of . In this article, we shall construct a set of 6k ? 2 disjoint (2,3)‐packings of order 6k + 4 with K_1,3 ∪ 3kK₂ or G₁ ∪ (3k ? 1)K₂ as their common leave for any integer k ≥ 1 with a few possible exceptions (G₁ is a special graph of order 6). Such a system can be used to construct perfect threshold schemes as noted by Schellenberg and Stinson ( 22 ). © 2006 Wiley Periodicals, Inc. J Combin Designs     

The links of 3‐dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5‐dimensional links of 3‐dimensional log‐canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ?⁴ such that the order of H₂(K) is 6k. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o‐minimal structure, then G is divisible. Furthermore, if G is defined in an o‐minimal expansion of a field, k ∈ ? and p_k : G → G is the definable map given by p_k (x ) = x^k for all x ∈ G , then we have |(p_k )^–1(x )| ≥ k^r for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Y‐rotation in k‐minimal quadrangulations on the projective plane

Let G be a quadrangulation on a surface, and let f be a face bounded by a 4‐cycle abcd. A face‐contraction of f is to identify a and c (or b and d) to eliminate f. We say that a simple quadrangulation G on the surface is k‐minimal if the length of a shortest essential cycle is k(≥3), but any face‐contraction in G breaks this property or the simplicity of the graph. In this article, we shall prove that for any fixed integer k≥3, any two k‐minimal quadrangulations on the projective plane can be transformed into each other by a sequence of Y‐rotations of vertices of degree 3, where a Y‐rotation of a vertex v of degree 3 is to remove three edges vv₁, vv₃, vv₅ in the hexagonal region consisting of three quadrilateral faces vv₁v₂v₃, vv₃v₄v₅, and vv₅v₆v₁, and to add three edges vv₂, vv₄, vv₆. Actually, every k‐minimal quadrangulation (k≥4) can be reduced to a (k?1)‐minimal quadrangulation by the operation called Möbius contraction, which is mentioned in Lemma 13. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 301–313, 2012     

Graph limits of random graphs from a subset of connected k‐trees

For any set Ω of non‐negative integers such that , we consider a random Ω‐k‐tree G_n,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that G_n,k, scaled by where H_k is the kth harmonic number and σ_Ω > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐k‐tree to an infinite but locally finite random Ω‐k‐tree G_∞,k.     

Continuous k‐to‐1 functions between complete graphs whose orders are of a different parity

A function between graphs is k‐to‐1 if each point in the codomain has precisely k pre‐images in the domain. Given two graphs, G and H, and an integer k≥1, and considering G and H as subsets of ?³, there may or may not be a k‐to‐1 continuous function (i.e. a k‐to‐1 map in the usual topological sense) from G onto H. In this paper we consider graphs G and H whose order is of a different parity and determine the even and odd values of k for which there exists a k‐to‐1 map from G onto H. We first consider k‐to‐1 maps from K_2r onto K_2s+1 and prove that for 1≤r≤s, (r, s)≠(1, 1), there is a continuous k‐to‐1 map for k even if and only if k≥2s and for k odd if and only if k≥?s?_o (where ?s?_o indicates the next odd integer greater than or equal to s). We then consider k‐to‐1 maps from K_2s+1 onto K_2s. We show that for 1≤r<s, such a map exists for even values of k if and only if k≥2s. We also prove that whatever the values of r and s are, no such k‐to‐1 map exists for odd values of k. To conclude, we give all triples (n, k, m) for which there is a k‐to‐1 map from K_n onto K_m in the case when n≤m. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 35–60, 2010.     

Multiple nontrivial solutions for resonant Neumann problems

We consider semilinear second order elliptic Neumann problems, which are resonant both at infinity (with respect to an eigenvalue λ_k, k ≥ 1) and at zero (with respect to the principal eigenvalue λ₀ = 0). Using techniques from Morse theory, combined with variational methods, we are able to show that the problem has at least four nontrivial smooth solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

P‐moment stability under small Gauss type random excitation of stochastic system

In this paper, the stochastic stability under small Gauss type random excitation is investigated theoretically and numerically. When p is larger than 0, the p‐moment stability theorem of stochastic models is proved by Lyapunov method, Ito isometry formula, matrix theory and so on. Then the application of p‐moment such as k‐order moment of the origin and k‐order moment of the center is introduced and analyzed. Finally, p‐moment stability of the power system is verified through the simulation example of a one machine and infinite bus system. Copyright © 2016 John Wiley & Sons, Ltd.     

Two‐factors each component of which contains a specified vertex

It is shown that if G is a graph of order n with minimum degree δ(G), then for any set of k specified vertices {v₁,v₂,…,v_k} ? V(G), there is a 2‐factor of G with precisely k cycles {C₁,C₂,…,C_k} such that v_i ∈ V(C_i) for (1 ≤ i ≤ k) if or 3k + 1 ≤ n ≤ 4k, or 4k ≤ n ≤ 6k ? 3,δ(G) ≥ 3k ? 1 or n ≥ 6k ? 3, . Examples are described that indicate this result is sharp. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 188–198, 2003     

A modified truncated singular value decomposition method for discrete ill‐posed problems

Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank‐k approximant, which we denote by A_k. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A_k. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A_k.Copyright © 2014 John Wiley & Sons, Ltd.     

6‐Star‐Coloring of Subcubic Graphs

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two adjacent vertices are assigned the same color) such that no path on four vertices is 2‐colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6‐star‐colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed (J Graph Theory 47(3) (2004), 140–153).     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

On the critical point‐arboricity graphs

In this paper, we study the critical point‐arboricity graphs. We prove two lower bounds for the number of edges of k‐critical point‐arboricity graphs. A theorem of Kronk is extended by proving that the point‐arboricity of a graph G embedded on a surface S with Euler genus g = 2, 5, 6 or g ≥ 10 is at most with equality holding iff G contains either K_2k?1 or K_2k?4 + C₅ as a subgraph. It is also proved that locally planar graphs have point‐arboricity ≤ 3 and that triangle‐free locally planar‐graphs have point‐arboricity ≤ 2. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 50–61, 2002     

Structure theorem for tournaments omitting N5

A finite tournament T is tight if the class of finite tournaments omitting T is well‐quasi‐ordered. We show here that a certain tournament N₅ on five vertices is tight. This is one of the main steps in an exact classification of the tight tournaments, as explained in [10]; the third and final step is carried out in [11]. The proof involves an encoding of the indecomposable tournaments omitting N₅ by a finite alphabet, followed by an application of Kruskal's Tree Theorem. This problem arises in model theory and in computational complexity in a more general form, which remains open: the problem is to give an effective criterion for a finite set {T₁,…,T_k} of finite tournaments to be tight in the sense that the class of all finite tournaments omitting each of T₁,…,T_k is well‐quasi‐ordered. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 165–192, 2003     

Extremes of deterministic sub‐sampled moving averages with heavy‐tailed innovations

Let {X_k}_k?1 be a strictly stationary time series. For a strictly increasing sampling function g:?→? define Y_k=X_g(k) as the deterministic sub‐sampled time series. In this paper, the extreme value theory of {Y_k} is studied when X_k has representation as a moving average driven by heavy‐tailed innovations. Under mild conditions, convergence results for a sequence of point processes based on {Y_k} are proved and extremal properties of the deterministic sub‐sampled time series are derived. In particular, we obtain the limiting distribution of the maximum and the corresponding extremal index. Copyright © 2003 John Wiley & Sons, Ltd.     

The diameter of randomly perturbed digraphs and some applications

The central observation of this paper is that if εn random arcs are added to any n‐node strongly connected digraph with bounded degree then the resulting graph has diameter 𝒪(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL‐complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ∼ 𝔻_n,ε/n we obtain a “smoothed” instance. We show that, with high probability, a log‐space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL ⫅̸ almost‐L then no heuristic can recognize similarly perturbed instances of (s,t)‐connectivity. Property Testing: A digraph is called k‐linked if, for every choice of 2k distinct vertices s₁,…,s_k,t₁,…,t_k, the graph contains k vertex disjoint paths joining s_r to t_r for r = 1,…,k. Recognizing k‐linked digraphs is NP‐complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k‐linked graphs with high probability, and rejects all graphs that are at least εn arcs away from being k‐linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Claw‐decompositions and tutte‐orientations

We conjecture that, for each tree T, there exists a natural number k_T such that the following holds: If G is a k_T‐edge‐connected graph such that |E(T)| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T. We prove that for T = K_1,3 (the claw), this holds if and only if there exists a (smallest) natural number k_t such that every k_t‐edge‐connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3‐flow conjecture says that k_t = 4. We prove the weaker statement that every 4$\lceil$ log n$\rceil$ ‐edge‐connected graph with n vertices has an edge‐decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge‐decomposition into claws. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 135–146, 2006     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011