On the use of graphs in discrete tomography

In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially solvable cases. We show how various classical techniques of operations research like matching, 2-SAT, network flows are applied to derive some of these results.      

积分第一中值定理中间点的一般渐近性质与求积公式

证明关于积分第一中值定理的中间点ξ的渐近性质的一般结果.而且,由此自然地推导出单节点数值积分公式.此求积公式具有高精度,还适于瑕积分的数值计算.     

钙钛矿型氧化物非常规铁电研究进展

钙钛矿型氧化物因具有丰富的磁性、铁电、力学和光学等诸多功能属性,在电子信息通信材料器件领域中有广阔的应用前景.在各种物理性质之中,铁电极化因其产生机制多样,并能与磁性和晶格应变相互耦合形成多铁性等特点,近十多年来一直被作为凝聚态物理研究的国际热点问题.与以自发极化作为初级序参量的常规铁电材料不同,非常规铁电材料中的铁电极化是被其他的序参量诱导而产生的.本综述围绕无机钙钛矿型氧化物非常规铁电体的研究进展进行了总结.回顾了该体系经典唯象理论和原子尺度的微观模型,有序排列的人工钙钛矿超晶格型结构,以及稀土正铁氧体单晶的反铁磁畴壁结构中非常规铁电的极化强度大小及其诱导机制,为系统理解非常规铁电提供了理论途径.     

Constraint Propagation in Graph Coloring

In this paper we propose a method for integrating constraint propagation algorithms into an optimization procedure for vertex coloring with the goal of finding improved lower bounds. The key point we address is how to get instances of Constraint Satisfaction Problems (CSPs) from a graph coloring problem in order to give rise to new lower bounds outperforming the maximum clique bound. More precisely, the algorithms presented have the common goal of finding CSPs in the graph for which infeasibility can be proven. This is achieved by means of constraint propagation techniques which allow the algorithms to eliminate inconsistencies in the CSPs by updating domains dynamically and rendering such infeasibilities explicit. At the end of this process we use the largest CSP for which it has not been possible to prove infeasibility as an input for an algorithm which enlarges such CSP to get a feasible coloring. We experimented with a set of middle-high density graphs with quite a large difference between the maximum clique and the chromatic number.     

Open colorings, the continuum and the second uncountable cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or ``open coloring axioms.' In particular it will be shown that the conjunction of two well-known axioms, and , implies that the size of the continuum is .

     

On Asymmetrically Decomposable Groups

Mathematical Notes -     

On the number of colorings of a snark minus an edge

For a given snark G and a given edge e of G, let ψ(G, e) denote the nonnegative integer such that for a cubic graph conformal to G – {e}, the number of Tait colorings with three given colors is 18 · ψ(G, e). If two snarks G₁ and G₂ are combined in certain well‐known simple ways to form a snark G, there are some connections between ψ (G₁, e₁), ψ (G₂, e₂), and ψ(G, e) for appropriate edges e₁, e₂, and e of G₁, G₂, and G. As a consequence, if j and k are each a nonnegative integer, then there exists a snark G with an edge e such that ψ(G, e) = 2^j · 3^k. © 2005 Wiley Periodicals, Inc.     

Locally restricted colorings

We consider the following problem: given suitable integers χ and p, what is the smallest value ρ such that, for any graph G with chromatic number χ and any vertex coloring of G with at most χ+p colors, there is a vertex v such that at least χ different colors occur within distance ρ of v? Let ρ(χ,p) be this value; we show in particular that ρ(χ,p)?⌈p/2⌉+1 for all χ,p. We give the exact value of ρ when p=0 or χ?3, and (χ,p)=(4,1) or (4,2).     

Acyclic list 7‐coloring of planar graphs

The acyclic list chromatic number of every planar graph is proved to be at most 7. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 83–90, 2002     

The maximum clique problem

In this paper we present a survey of results concerning algorithms, complexity, and applications of the maximum clique problem. We discuss enumerative and exact algorithms, heuristics, and a variety of other proposed methods. An up to date bibliography on the maximum clique and related problems is also provided.