最大k—一致超图

本文刻划直径为ｄ 的最大边数的ｋ－一致超图的结构,推广了Ｏｒｅ 的一个结果     

The poset of hypergraph quasirandomness

Chung and Graham began the systematic study of k‐uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung‐Graham‐Wilson on quasirandom graphs. One feature that became apparent in the early work on k‐uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,762–800, 2015     

On k -diameter of k -connected graphs

Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d _k (G) for which between any two vertices in G there are at least k internally vertex-disjoint paths of length at most d _k (G). For a fixed positive integer d, some conditions to insure d _k (G)⩽d are given in this paper. In particular, if d⩾3 and the sum of degrees of any s (s=2 or 3) nonadjacent vertices is at least n+(s−1)k+1−d, then d _k (G)⩽d. Furthermore, these conditions are sharp and the upper bound d of k-diameter is best possible. Supported by NNSF of China (19971086).     

Rapid mixing of hypergraph independent sets

We prove that the mixing time of the Glauber dynamics for sampling independent sets on n‐vertex k‐uniform hypergraphs is when the maximum degree Δ satisfies Δ ≤ c2^k/2, improving on the previous bound Bordewich and co‐workers of Δ ≤ k ? 2. This result brings the algorithmic bound to within a constant factor of the hardness bound of Bezakova and co‐workers which showed that it is NP‐hard to approximately count independent sets on hypergraphs when Δ ≥ 5·2^k/2.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

Point Sets with Many k -Sets

For any n , k , n\geq 2k>0 , we construct a set of n points in the plane with k -sets. This improves the bounds of Erdős, Lovász, et al. As a consequence, we also improve the lower bound for the number of halving hyperplanes in higher dimensions. Received September 10, 1999, and in revised form January 27, 2000.     

The quasi‐randomness of hypergraph cut properties

Let satisfy and suppose a k‐uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets of sizes , the number of edges intersecting is (asymptotically) the number one would expect to find in a random k‐uniform hypergraph. Can we then infer that H is quasi‐random? We show that the answer is negative if and only if . This resolves an open problem raised in 1991 by Chung and Graham [J AMS 4 (1991), 151–196]. While hypergraphs satisfying the property corresponding to are not necessarily quasi‐random, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasi‐random hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

图带宽和与其对偶超图带宽和的关系

设H=（E1,E2,…,Em）是集合X上的一个超图,一个1-1映射f:X→{1,2,…,|X|}称为H的一个标号,对H的任一标号f,BS（H,f）=∑(E∈H)max{|f(u)-f(v)|;u,v∈E}称为超图H的关于标号f的带宽和BS(H)=min{BS(H,f)|f是超图H的标号|}称为H的带宽和．论文研究图带宽和与其对偶超图的带宽和这两个参数间的关系．     

On products of k atoms

Let H be an atomic monoid. For k ? \Bbb Nk \in {\Bbb N} let V_k (H){\cal V}_k (H) denote the set of all m ? \Bbb Nm \in {\Bbb N} with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets V_k (H){\cal V}_k (H) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every k ? \Bbb Nk \in {\Bbb N}, max V_2k+1 (H) = k |G|+ 1{\cal V}_{2k+1} (H) = k \vert G\vert + 1 which settles Problem 38 in [4].     

(k,+)-distance-hereditary graphs

In this work we introduce, characterize, and provide algorithmic results for (k,+)-distance-hereditary graphs, k0. These graphs can be used to model interconnection networks with desirable connectivity properties; a network modeled as a (k,+)-distance-hereditary graph can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is bounded by the distance in the non-faulty graph plus an integer constant k. The class of all these graphs is denoted by DH(k,+). By varying the parameter k, classes DH(k,+) include all graphs and form a hierarchy that represents a parametric extension of the well-known class of distance-hereditary graphs.     

Faster algorithms for string matching with k mismatches

The string matching with mismatches problem is that of finding the number of mismatches between a pattern P of length m and every length m substring of the text T. Currently, the fastest algorithms for this problem are the following. The Galil–Giancarlo algorithm finds all locations where the pattern has at most k errors (where k is part of the input) in time O(nk). The Abrahamson algorithm finds the number of mismatches at every location in time . We present an algorithm that is faster than both. Our algorithm finds all locations where the pattern has at most k errors in time . We also show an algorithm that solves the above problem in time O((n+(nk³)/m)logk).     

Superconnectivity of regular graphs with small diameter

A graph is superconnected, for short super-κ, if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any r-regular graph of diameter D and odd girth g that if D≤g−2, then the graph is super-κ when g≥5 and a complete graph otherwise.     

Existence of solutions to (k, n − k − 2) discrete boundary value problems

We provide general existence criteria to solutions for a class of higher-order discrete boundary value problems. Our results supplement as well as improve several recent results established in the literature.     

A simplex variant solving an m × d linear program in O(min(m, d) expected number of pivot steps

We present a variant of the Simplex method which requires on the average at most 2 (min(m, d) + 1)² pivots to solve the linear program min c^T, Ax ≥ b, x ≥ 0 with A R^m×d. The underlying probabilistic distribution is assumed to be invariant under inverting the sense of any subset of the inequalities. In particular, this implies that under Smale's spherically symmetric model this variant requires an average of no more than 2(d + 1)² pivots, independent of m, where d ≤ m.     

On the largest kth eigenvalues of trees with n≡0 (mod k)

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue of trees with n vertices. In 1995, Jia-yu Shao gave complete solutions for the cases k=2,3,4,5,6 and provided some necessary conditions for extremal trees in general cases (cf. [Linear Algebra Appl. 221 (1995) 131]). We further improve Shao's necessary condition in this paper, which can be seen as the continuation of [Linear Algebra Appl. 221 (1995) 131].     

An SDP randomized approximation algorithm for max hypergraph cut with limited unbalance

We consider the design of semidefinite programming(SDP) based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance(MHC-LU): Find a partition of the vertices of a weighted hypergraph H =(V, E) into two subsets V1, V2 with ‖V2|- |V1‖ u for some given u and maximizing the total weight of the edges meeting both V1 and V2. The problem MHC-LU generalizes several other combinatorial optimization problems including Max Cut, Max Cut with Limited Unbalance(MC-LU), Max Set Splitting,Max Ek-Set Splitting and Max Hypergraph Bisection. By generalizing several earlier ideas, we present an SDP randomized approximation algorithm for MHC-LU with guaranteed worst-case performance ratios for various unbalance parameters τ = u/|V|. We also give the worst-case performance ratio of the SDP-algorithm for approximating MHC-LU regardless of the value of τ. Our strengthened SDP relaxation and rounding method improve a result of Ageev and Sviridenko(2000) on Max Hypergraph Bisection(MHC-LU with u = 0), and results of Andersson and Engebretsen(1999), Gaur and Krishnamurti(2001) and Zhang et al.(2004) on Max Set Splitting(MHC-LU with u = |V|). Furthermore, our new formula for the performance ratio by a tighter analysis compared with that in Galbiati and Maffioli(2007) is responsible for the improvement of a result of Galbiati and Maffioli(2007) on MC-LU for some range of τ.     

On L(d,1)-labelings of graphs

Given a graph G and a positive integer d, an L(d,1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u)−f(v)|d; if u and v are not adjacent but there is a two-edge path between them, then |f(u)−f(v)|1. The L(d,1)-number of G, λ_d(G), is defined as the minimum m such that there is an L(d,1)-labeling f of G with f(V){0,1,2,…,m}. Motivated by the channel assignment problem introduced by Hale (Proc. IEEE 68 (1980) 1497–1514), the L(2,1)-labeling and the L(1,1)-labeling (as d=2 and 1, respectively) have been studied extensively in the past decade. This article extends the study to all positive integers d. We prove that λ_d(G)Δ²+(d−1)Δ for any graph G with maximum degree Δ. Different lower and upper bounds of λ_d(G) for some families of graphs including trees and chordal graphs are presented. In particular, we show that the lower and the upper bounds for trees are both attainable, and the upper bound for chordal graphs can be improved for several subclasses of chordal graphs.