Local and Union Page Numbers

Abstract

We introduce the novel concepts of local and union book embeddings, and, as the corresponding graph parameters, the local page number pn (G) and the union page number pn u (G). Both parameters are relaxations of the classical page number pn(G), and for every graph G we have pn (G) pn u (G) pn(G). While for pn(G) one minimizes the total number of pages in a book embedding of G, for pn (G) we instead minimize the number of pages incident to any one vertex, and for pn u (G) we instead minimize the size of a partition of G with each part being a vertex-disjoint union of crossing-free subgraphs. While pn (G) and pn u (G) are always within a multiplicative factor of 4, there is no bound on the classical page number pn(G) in terms of pn (G) or pn u (G). We show that local and union page numbers are closer related to the graph's density, while for the classical page number the graph's global structure can play a much more decisive role. We introduce tools to investigate local and union book embeddings in exemplary considerations of the class of all planar graphs and the class of graphs of tree-width k.As an incentive to pursue research in this new direction, we offer a list of intriguing open problems.

The Local Queue Number of Graphs with Bounded Treewidth

“…The latter might offer a way to support conjectured upper bounds on the classical queue number. We remark that analogously to the local queue number considered here, we recently introduced [17] the local page number as a weaker version of the classical page number.…”

“…The proofs in this section follow the proofs of the analogous theorems on the local page number in [17]. We start with a proof of Theorem 2 which claims that the local queue number is tied to the maximum average degree.…”

“…The local version of page numbers was introduced and investigated in [17]. The local page number of any graph is always near its maximum average degree, while the classical page number can be arbitrarily far off: For any d 3, there are n-vertex graphs with local page number at most d + 2 but page number Ω( √ dn 1/2−1/d ).…”

“…In addition, the local queue number is tied to the maximum average degree, which is defined as mad(G) = max{2|E(H)|/|V (H)| : H ⊆ G, H = ∅}. Both results are derived from the analogous results for local page numbers [17], which is why we omit the proofs here. They can be found in Appendix A. Theorem 2 also implies that the local queue number is tied to the local page number, which is conjectured for the classical page number and classical queue number.…”

“…Surprisingly, the union queue number is tied to the local queue number and we have a linear upper bound on the union queue number of k-trees. We refer to [17] for an analogous proof for local and union page numbers. This observation has interesting consequences for queue layouts of k-trees.…”