Linear Layouts of Complete Graphs

TL;DR

This paper investigates various layout parameters of complete graphs, providing tight bounds and exact values for the union page, union queue, local page, and local queue numbers, advancing understanding of graph layout complexities.

Contribution

It establishes new bounds and exact results for four key layout parameters of complete graphs, including the local page number as approximately n/3.

Findings

Local page number of K_n is approximately n/3.

Local and union queue numbers of K_n are about (1-1/√2)n.

Union page number of K_n is between n/3 and 4n/9.

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) $.