Linear Layouts of Complete Graphs

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 $4n/9 + O(1)$.