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 ${\rm pn}_\ell(G)$ and the union page number ${\rm pn}_u(G)$. Both parameters are relaxations of the classical page number ${\rm pn}(G)$, and for every graph $G$ we have ${\rm pn}_\ell(G) \leq {\rm pn}_u(G) \leq {\rm pn}(G)$. While for ${\rm pn}(G)$ one minimizes the total number of pages in a book embedding of $G$, for ${\rm pn}_\ell(G)$ we instead minimize the number of pages incident to any one vertex, and for ${\rm pn}_u(G)$ we instead minimize the size of a partition of $G$ with each part being a vertex-disjoint union of crossing-free subgraphs. While ${\rm pn}_\ell(G)$ and ${\rm pn}_u(G)$ are always within a multiplicative factor of $4$, there is no bound on the classical page number ${\rm pn}(G)$ in terms of ${\rm pn}_\ell(G)$ or ${\rm pn}_u(G)$. 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.