双色图直线嵌入的泛集
Universal Sets for Straight-Line Embeddings of Bicolored Graphs
摘要：
一个n点集S对一个n阶图G是2-色泛的，如果对G的每一个真2-染色和对S的每一个具有与G相同色类大小的2-染色，G是直线可嵌入到S上的.我们表明，所谓的双链是2色泛的路径，如果两个链中的每一个包含至少五分之一的所有点，但如果其中一个链是超过约28倍的长度比其他。如果G的色类的大小相差不超过1，则G的2-着色是公平的。一个二部图是公平的，如果它允许一个公平的真染色。我们研究的情况下，S是双链的链大小相差最多1和G是一个公平的二部图。我们证明了这个S不是2色泛的，如果G不是一个森林的毛毛虫，它是2色泛的公平毛毛虫最多有一半的非叶顶点。我们还表明，如果这个S是公平的2色，然后公平适当的2色森林的恒星可以嵌入它。
英文摘要：
A set S of n points is 2-color universal for a graph G on n vertices if for every proper 2-coloring of G and for every 2-coloring of S with the same sizes of color classes as G has, G is straight-line embeddable on S. We show that the so-called double chain is 2-color universal for paths if each of the two chains contains at least one fifth of all the points, but not if one of the chains is more than approximately 28 times longer than the other. A 2-coloring of G is equitable if the sizes of the color classes differ by at most 1. A bipartite graph is equitable if it admits an equitable proper coloring. We study the case when S is the double-chain with chain sizes differing by at most 1 and G is an equitable bipartite graph. We prove that this S is not 2-color universal if G is not a forest of caterpillars and that it is 2-color universal for equitable caterpillars with at most one half non-leaf vertices. We also show that if this S is equitably 2-colored, then equitably properly 2-colored forests of stars can be embedded on it.
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