Linear Layouts of Complete Graphs

Abstract

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) $.