# Local and Union Page Numbers

## TL;DR

This paper introduces local and union page numbers as new graph parameters that relax classical page number, explores their properties, relationships with graph density, and their behavior in planar and bounded tree-width graphs.

## Contribution

It defines and studies local and union page numbers, establishing their bounds, relationships, and differences from classical page number, and provides tools and open problems for further research.

## Key findings

- Local and union page numbers are always within a factor of 4.
- No bound exists on classical page number in terms of local or union page numbers.
- Local and union page numbers relate closely to graph density.

## 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.