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 \(n/3 \pm \mathcal {O}(1) \), while its local and union queue number is \((1-1/\sqrt{2})n \pm \mathcal {O}(1) \). The union page number of \( K_n \) is between \( n/3 - \mathcal {O}(1) \) and \( 4n/9 + \mathcal {O}(1) \).
S. Felsner—Partially supported by DFG grant FE 340/13-1.
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Notes
- 1.
Indeed, for each n between 56 and 294 and corresponding k, one of the following two cases applies. First, there is an even \( n' \geqslant n \) such that \( k' = \lceil (1-1/\sqrt{2})(n'+1) \rceil \) is even, \( 3 n' \geqslant 10 k' \) holds, and we have \( \mathrm {qn}_\ell (K_n) \leqslant \mathrm {qn}_\ell (K_{n'}) \leqslant k' + 9 \leqslant k + 11 \), respectively \( {\mathrm {qn}}_{\mathrm {u}}(K_n) \leqslant {\mathrm {qn}}_{\mathrm {u}}(K_{n'}) \leqslant k' + 40 \leqslant k + 42 \). Or second, there is an even \( n' \) such that \( n - 4 \leqslant n' \leqslant n \), \( k' = \lceil (1-1/\sqrt{2})(n'+1) \rceil \) is even, \( 3 n' \geqslant 10 k' \) holds, and the queue layout we obtain for \( K_{n'} \) can be augmented to a queue layout of \( K_n \) matching the desired bounds. To do so, we attach at most two of the additional \( n' - n \leqslant 4 \) vertices to the left and at most two to the right and use two additional queues to cover the new edges. We then observe that \( k' + 2 \leqslant k \), which gives the desired bounds for \( K_n \).
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The first, third and fourth author would like to thank the organizers and all participants of the Seventh Annual Workshop on Geometry and Graphs in Barbados, where part of this research was carried out.
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Felsner, S., Merker, L., Ueckerdt, T., Valtr, P. (2021). Linear Layouts of Complete Graphs. In: Purchase, H.C., Rutter, I. (eds) Graph Drawing and Network Visualization. GD 2021. Lecture Notes in Computer Science(), vol 12868. Springer, Cham. https://doi.org/10.1007/978-3-030-92931-2_19
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