完全图的线性布局
Linear Layouts of Complete Graphs
摘要：
关于图的顶点排序的页面（队列）是使得没有两个边交叉（嵌套）的边的集合，即，他们的结局是以ABAB模式（ABBA模式）排序的。联合页面（联合队列）是页面（队列）的顶点不相交联合。图的联合页号（联合队列号）是最小的$ k $，使得存在顶点排序和将边划分为$ k $联合页（联合队列）。局部页号（局部队列号）是最小的$ k $，对于该最小的$ k $，存在顶点排序和将边划分为页（队列），使得每个顶点在至多$ k $页（队列）中具有关联边。我们提出了这四个参数的上限和下界的完全图$ K_n $上$ n $顶点。在三种情况下，我们得到了精确的结果，直到一个加法常数。具体地，$ K_n $的本地页号是$ n/3 \pm O(1) $，而其本地和联合队列号是$ (1-1/\sqrt{2})n \pm O(1) $。$ K_n $的联合页码介于$ n/3 - O(1) $和$ 4n/9 + O(1) $之间。
英文摘要：
A page (queue) with respect to a vertex ordering of a graph is a set of edges such that no two edges cross (nest), i.e., have their endpoints ordered in an ABAB-pattern (ABBA-pattern). A union page (union queue) is a vertex-disjoint union of pages (queues). The union page number (union queue number) of a graph is the smallest $ k $ such that there is a vertex ordering and a partition of the edges into $ k $ union pages (union queues). The local page number (local queue number) is the smallest $ k $ for which there is a vertex ordering and a partition of the edges into pages (queues) such that each vertex has incident edges in at most $ k $ pages (queues). We present upper and lower bounds on these four parameters for the complete graph $ K_n $ on $ n $ vertices. In three cases we obtain the exact result up to an additive constant. In particular, the local page number of $ K_n $ is $ n/3 \pm O(1) $, while its local and union queue number is $ (1-1/\sqrt{2})n \pm O(1) $. The union page number of $ K_n $ is between $ n/3 - O(1) $ and $ 4n/9 + O(1) $.
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