Linear Layouts of Complete Graphs
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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 ± O(1), while its local and union queue number is (1-1/√(2))n ± O(1). The union page number of K_n is between n/3 - O(1) and 4n/9 + O(1).
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