分层空间图的性质
Properties of leveled spatial graphs
AI总结：
本文研究空间图的分层嵌入性质，引入层数和哈密顿层数两个新不变量以度量图与平面图的距离，并刻画低层数图，确定完全图及完全二部图的层数值。
摘要：
我们研究了空间图具有分层嵌入的性质，并刻画了具有该性质的抽象图。我们证明所有分层嵌入都是自由的，并比较了分层嵌入与面板（也称为平坦）嵌入。我们引入了两个新的图不变量：层数，这是对允许分层嵌入的图的一个不变量；以及哈密顿层数，这是对哈密顿图的一个不变量。这些不变量提供了一种度量，用于衡量一个图距离平面图有多远。我们研究了（哈密顿）层数与其它图不变量之间的关系，这些不变量最小化图在平面子图中的分解，即图的厚度和书厚度。我们刻画了具有低层数的图，并确定了完全图和完全二部图的层数与哈密顿层数。
英文摘要：
We investigate the property of a spatial graph of having a leveled embedding and characterize the abstract graphs with this property. We show that all leveled embeddings are free and we compare leveled and paneled (also known as flat) embeddings. Two new graph invariants are introduced: the level number, an invariant for graphs that admit a leveled embedding, and the Hamiltonian level number, an invariant for Hamiltonian graphs. These invariants provide a measure on how far a graph is from being planar. We study the relation between the (Hamiltonian) level number and other graph invariants that minimize the decomposition of a graph in planar subgraphs, namely the thickness and the book thickness of a graph. We characterize graphs with low level number and determine both the level number and the Hamiltonian level number of complete graphs and of complete bipartite graphs.
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