Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

A Note on q‐Covering Designs in PG(5, q)

A construction of q‐covering designs in PG(5, q) is given, providing an improvement on the upper bound of the q‐covering number .     

On the sum minimization version of the online bin covering problem

Given a set of m identical bins of size 1, the online input consists of a (potentially infinite) stream of items in (0,1]. Each item is to be assigned to a bin upon arrival. The goal is to cover all bins, that is, to reach a situation where a total size of items of at least 1 is assigned to each bin. The cost of an algorithm is the sum of all used items at the moment when the goal is first fulfilled. We consider three variants of the problem, the online problem, where there is no restriction of the input items, and the two semi-online models, where the items arrive sorted by size, that is, either by non-decreasing size or by non-increasing size. The offline problem is considered as well.     

On the relations of graph parameters and its total parameters

In this paper we get some relations between α（G）, α＇（G）, β（G）, β＇（G） and αT（G）, βT（G）. And all bounds in these relations are best possible, where α（G）, α＇（G）,/3（G）, β（G）, αT（G） and βT（G） are the covering number, edge-covering number, independent number, edge-independent number （or matching number）, total covering number and total independent number, respectively.     

用随机小区间覆盖圆周(英文)

本文考虑随机区间In=ωn+(-en/2,en/2)(mod 1).利用文献[7]中所介绍的无处不在系统,证明了圆周上由被无穷次覆盖的点构成的集合的Hausdorff维数几乎必然等于min{1,lim sup n→∞ log n/-log en},推广了文献[4]中的结果.     

Computing the minimal covering set

We present the first polynomial-time algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set–the minimal upward covering set and the minimal downward covering set–unless P equals NP. Finally, we observe a strong relationship between von Neumann–Morgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.     

The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids Γ, the E-theoretic descent functor at the C^∗-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. Section 2 shows that Γ-actions on a C₀(X)-algebra B, where X is the unit space of Γ, can be usefully formulated in terms of an action on the associated bundle B^?. Section 3 shows that the functor B→C^∗(Γ,B) is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ, the main technical difficulty involved being that of finding a Γ-algebra that plays the role of Cbcont(T,B) in the group case.     

VC‐dimension on manifolds: a first approach

The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd.     

单位圆内有限级拟亚纯映射在Borel半径上的充满圆

应用覆盖曲面的几何方法,对于单位圆内有限正级的K-拟亚纯映射在其Borel半径上的性质进行了研究,用比较简单的方法证明了单位圆内有限正级K-拟亚纯映射在其Borel半径上一定存在充满圆序列,推广了Rauch A的结果.     

罗尔中值定理的一些新证法(英文)

给出了罗尔微分中值定理的三种新的证明方法,其中第二种很简便的方法仅依赖于大家熟知的Heine-Borel有限覆盖定理.由此可见罗尔微分中值定理可以是实数的完备性的直接推论.