Solving some NP-complete problems using split decomposition

We show how to use the split decomposition to solve some NP-hard optimization problems on graphs. We give algorithms for clique problem and domination-type problems. Our main result is an algorithm to compute a coloration of a graph using its split decomposition. Finally we show that the clique-width of a graph is bounded if and only if the clique-width of each representative graph in its split decomposition is bounded.     

A local structure theorem on 5‐connected graphs

An edge of a 5‐connected graph is said to be contractible if the contraction of the edge results in a 5‐connected graph. Let x be a vertex of a 5‐connected graph. We prove that if there are no contractible edges whose distance from x is two or less, then either there are two triangles with x in common each of which has a distinct degree five vertex other than x, or there is a specified structure called a K₄^?‐configuration with center x. As a corollary, we show that if a 5‐connected graph on n vertices has no contractible edges, then it has 2n/5 vertices of degree 5. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 99–129, 2009     

Batch processing with interval graph compatibilities between tasks

We analyze batch-scheduling problems that arise in connection with certain industrial applications. The models concern processing on a single max-batch machine with the additional feature that the tasks of the same batch have to be compatible. Compatibility is a symmetric binary relation—the compatible pairs are described with an undirected “compatibility graph”, which is often an interval graph according to some natural practical conditions that we present. We consider several models with varying batch capacities, processing times or compatibility graphs. We summarize known results, and present a min-max formula and polynomial time algorithms.     

Distinguishing colorings of Cartesian products of complete graphs

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K_s,t which admits only the identity automorphism. In particular, this allows us to determine the distinguishing number of the Cartesian product of complete graphs.     

Game coloring the Cartesian product of graphs

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G^* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G^* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G□G′ has game chromatic number at most k(k + m ? 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008     

图的倍图与补倍图

计算机科学数据库的关系中遇到了可归为倍图或补倍图的参数和哈密顿圈的问题．对简单图C，如果V（D（G））：V（G）∪V（G′）E（D（G））=E（C）∪E（C″）U{vivj′｜vi∈V（G），Vj′∈V（G′）且vivj∈E（G））那么，称D（C）是C的倍图，如果V（D（G））=V（C）∪V（G′），E（D（C））：E（C）∪E（G′）∪{vivj′}vi∈V（G），vj′∈V（G’）and vivj∈（G）），称D（C）是G的补倍图，这里G′是G的拷贝．本文研究了D（G）和D的色数，边色数，欧拉性，哈密顿性和提出了D（G） 的边色数是D（G）的最大度等公开问题．     

Reverse Mathematics and Grundy colorings of graphs

The relationship of Grundy and chromatic numbers of graphs in the context of Reverse Mathematics is investi‐gated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The analytical solution and the complete electronic structure of quantum well

With a detailed analysis of the Bloch wave properties of one-dimensional infinite crystal, we find the exact analytical solution of the quantum confinement states in quantum well with infinite barrier. Based on the analytical solution, we calculate the complete electronic structure of quantum well with arbitrary geometries. The calculation results demonstrate that the transitions only occur between specific pairs of sub-bands. It also shows that the band edge state plays an insignificant role in the optical properties, although it is the most distinguishing feature of the quantum well.     

Smaller planar triangle-free graphs that are not 3-list-colorable

In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices. We present two more graphs with these properties. The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors.     

一端固支一端简支变厚度梁的弹性力学解

研究了一端固支另一端简支连续变厚度梁在静力荷载作用下的应力和位移分布.通过引入单位脉冲函数和Dirae函数,将固支边等效为简支边与未知水平反力的叠加,利用平面应力问题的基本方程,导出满足控制微分方程及左右两端边界条件的位移函数的一般解,对上下表面的边界方程作Fourier级数展开,结合固支边位移为O的条件确定待定系数,得到的解是高精度的.数值结果与商业有限元软件ANSYS进行了比较,显示出很好的精度.