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The linear choosability numberΛ~l of a graph G is the minimum number k for which G is hnear vertex k-list colorable.

图G的线性点可选数是使得G是线性点k-列表可选的最小正整数，记为Λ~l。

Yap proved that the planar graph with maximum degree △≥ 13 is equitably m—colorable for any m ≥△.

Yap和Zhang[10]证明了：最大度Δ≥13的任意平面图G，对任意整数m≥Δ，都存在均匀m-染色。

A graph G is acyclically L-list colorable if for a given list assignment L there is an acyclic coloringπof the vertices such thatπ∈L.

若G有一个正常的无圈染色π，使得对每一个顶点v∈V，都有π∈L，则称G是无圈L-可染的或者称π是G的一个无圈L-染色。

Basis on this result, Chen, Lih and Wu put forth the following conjecture:Conjecture 2 For any connected graph G, except K_m, C_(2m+1) and K_(2m+1,2m+1) for all m ≥ 0,then G is equitably △—colorable .

基于这一结果，Chen，Lih和Wu提出了以下猜想：猜想2：如果图G是一个既不是完全图，奇圈又不是完全二部图K_(2m+1，2m+1)，m≥0的连通图，那么G存在均匀Δ-染色。

If G is acyclically L-list colorable for any list assignment with |L|≥κfor all v∈V,then G is acyclicallyκ-choosable.The acyclic list chromatic number,denoted byχ_a~l,of a graph G is the smallest integerκsuch that G is acyclicallyκ-choosable.

若对满足|L|≥κ，v∈V的色列表L，G都是无圈L-可染的，那么称G是无圈κ-可选择的。G的无圈列表色数χ_a~l是使得G是无圈κ-可选择的最小的非负整数κ。

This graph must have capability of zoom in, zoom out, and colorization.

该图必须有变焦能力，缩小，和着色。

Columnar vertical line graph is based on the green line and the composition of the green cylinders.

柱状垂直线图是以绿色线与绿色柱体构成的。

Each point on this graph denotes a combination of a price and a quantity.

图中每一点表示价格和数量的组合。

Today, at 23, he has published over 100 academic papers in fields as diverse as computational geometry, combinatorial game theory, data structures and graph theory. Along with his interest in folding, Dr. Demaine is also an expert in algorithms.

今日二十三岁的他，已出版了超过一百份学术报告分别在多种不同的领域如计算几何学、组合博弈论、资料结构以及图论，他除了对摺纸感兴趣，迪铭博士也是位演算法专家。

Other semirings arise naturally in such diverse areas of mathematics as combinatorics, functional analysis, topology, graph theory, Euclidean geometry, probability theory, commutative and noncommutative ring theory, optimization theory, discrete event dynamical systems, automata theory, formal language theory and the mathematical modeling of quantum physics and parallel computation systems.

在许多不同的数学领域存在着其它形式的半环，比如组合学，泛函分析，拓扑学，图论，欧式几何，概率论，交换和非交换环，优化论，离散动力系统，自动化理论，形式语言理论，量子物理的数据模型和并行计算系统。

In the negative and interrogative forms, of course, this is identical to the non-emphatic forms.

。但是，在否定句或疑问句里，这种带有"do"的方法表达的效果却没有什么强调的意思。

Go down on one's knees;kneel down

屈膝跪下。。。下跪祈祷