A linear embedding of a graph is a tuple where is a total ordering111We define as a linear ordering. However, in a few places we shall think of as a cyclic ordering. This is legitimate as we are interested in crossing edges only, and these are preserved under cyclic shifts. of the vertex set and is a partition of the edge set . The ordering is sometimes called the spine ordering, and each part of is called a page. For a given spine ordering , two edges with and are said to be crossing if . A linear embedding is a book embedding if for any two edges and in we have
So Eq. 1 simply states that no two edges in the same page are crossing, or equivalently, any two crossing edges are assigned to distinct pages in .
Book embeddings were introduced by Ollmann  as well as Bernhart and Kainen , see also . Besides their apparent applications in real-world problems (see e.g. [26, 8] and the numerous references in ), book embeddings enjoy steady popularity in graph theory; see for example [30, 11, 16, 27, 21, 33], just to name a few. In most cases (also including the generalizations for directed graphs  or pages with limited crossings ), one seeks to find a book embedding with as few pages as possible for given graph . In particular, is a -page book embedding if , and the page number of , denoted by , is the smallest for which we can find a -page book embedding of . (We remark that is sometimes also called the book thickness  or stack number  of .)
As the main contribution of the present paper, we propose two relaxations of the page number parameter: The local page number and the union page number . We initiate the study of these parameters by comparing , , and for graphs in some natural graph classes, such as planar graphs (c.f. Section 3), graphs of bounded density (c.f. Section 2), and graphs of bounded tree-width (c.f. Section 4). Besides these bounds, a (perhaps not surprising) result showing computational hardness (c.f. Theorem 1.4), and a few structural observations, we also give some intriguing open problems at the end of the paper in Section 5.
Before listing our specific results in Section 1.1 below, let us define and motivate the novel parameters local and union page numbers.
Local Page Numbers. For a book embedding of graph and a vertex , let us denote by the subset of pages that contain at least one edge incident to . Then we define:
A book embedding is -local if for each , i.e., each vertex has incident edges on at most pages.
The local page number, denoted by , is the smallest for which we can find a -local book embedding of .
Thus, we seek to find a book embedding with any number of pages (possibly more than ), but with no vertex having incident edges on more than of these pages. As each -page book embedding is a -local book embedding,
However, can be strictly smaller than . For example, and both have page number and local page number . As illustrated in Fig. 1, admits a -local -page book embedding, i.e., this book embedding simultaneously certifies and . In the left of Fig. 1 we have a -local -page book embedding of (when the three orange/thick edges are put into three separate pages). So here, the introduction of “extra” pages, additionally to the necessary pages in every book embedding of , allowed us to actually reduce the maximum number of pages incident to any one vertex from to . And for some graphs with , in fact all -local book embeddings have more than pages.
Union Page Numbers. For a linear embedding (so not necessarily a book embedding) of graph and a page , let us denote by the subgraph of on all edges in and all vertices with some incident edge in . Then we define:
A linear embedding is a union embedding if is a (-page) book embedding for each connected component of and each , i.e., each connected component of each page is crossing-free.
The union page number, denoted by , is the smallest for which we can find a -page union embedding of .
In other words, in a union embedding, each page is the vertex-disjoint union of crossing-free graphs; hence the name “union page number”. So we allow crossing edges on a single page , as long as these are contained in different components of . For the union page number we minimize the number of pages, just like for the classical page number .
Again, each -page book embedding is also a -page union embedding, giving . Moreover, each -page union embedding can be transformed into a -local book embedding by putting each component of each page onto a separate page, giving . Summarizing,
Consider again the linear embedding of in the left of Fig. 1, but this time put all three orange/thick edges on the same page . These edges are pairwise crossing, so this is not a book embedding. However these edges lie in separate connected components of , so this is a union embedding. As we found a -page union embedding of , we see .
Comparing union and local page numbers, we have that can be strictly smaller than . For example, we have already seen in Fig. 1 that , and we claim that . Indeed, for the cyclic spine ordering and pages we may assume by symmetry that and . As each connected component of and is crossing-free, and are in distinct components in both page and page , leaving no way to assign the edge .
Motivation. Local and union page numbers are motivated by local and union covering numbers as introduced by Knauer and the second author . In order to give a brief summary of the covering number framework, consider a graph and a graph class . An injective -cover of is a set of subgraphs222In a general -cover one considers graph homomorphisms from graphs in into . However, we consider here only injective -covers, which is equivalent to considering subgraphs of . of such that and for . In other words, is covered by (is the union of) some (possibly isomorphic) graphs from . Moreover, let denote the class of all finite vertex-disjoint unions of graphs in , meaning that if and only if is the vertex-disjoint union of some number of graphs in .
The global -covering number of , denoted by , is the smallest such that there exists an injective -cover of of size , i.e., using graphs in . The union -covering number of , denoted by , is the smallest such that there exists an injective -cover of of size , i.e., using vertex-disjoint unions of graphs in . The local -covering number of , denoted by , is the smallest such that there exists an injective -cover of in which every vertex of is contained in at most graphs of the cover, i.e., using any number of graphs from but with no vertex of being contained in more than of these.333The covering number framework includes a fourth covering number, the folded -covering number of , which we omit here, so as not to congest the discussion.
Many graph parameters (including arboricities, thickness parameters, variants of chromatic numbers, several Ramsey numbers, and interval representations) are -covering numbers of a certain type and for a certain graph class . Moreover, recently the global-union-local framework was extended to settings that do not directly concern graph covers, such as the local and union boxicity , and the local dimension of posets , which has stimulated research drastically [20, 12, 28, 18, 7, 2]. Our proposed local and union page numbers naturally arise from the covering number framework by using ordered graphs and ordered subgraphs in the above definitions and taking to be the class of all crossing-free ordered graphs.
Particularly the local page number might be very useful in applications. For example, oftentimes the spine ordering of is already given from the problem formulation (by time stamps, geographic positions or a genetic sequence). Then the edges of model some kind of connections and classical book embeddings are used to distribute the connections to machines that can process sets of connections that satisfy the LIFO (last-in-first-out) property. Local book embeddings could be used to model situations in which the total number of machines is not the scarce resource but rather the number of machines working on the same element, i.e., vertex. Imagine for example limited capacity at each element in terms of computing power (as for cell phones) or simply spatial restrictions (as for genes). This kind of task is precisely modeled by local book embeddings and the local page numbers.
1.1 Our Contribution
First, we show that the new parameters and can be arbitrarily smaller than the classical page number , while local and union page number are always at most a multiplicative factor of apart.
For any and infinitely many values of , there exist -vertex graphs with
In contrast, for every graph we have .
While for every planar graph we have , it is not known whether there is a planar graph with . The best known lower bound was given by Bernhart and Kainen , who presented a planar graph with . That very graph satisfies , but we can augment it to a planar graph with local page number .
There is a planar graph with .
For every there is a graph of tree-width with .
Finally, it is known that if and only if is a subgraph of a planar Hamiltonian graph . Hence, it follows from  that deciding is NP-complete, which easily generalizes to for each . (Since is equivalent to being outerplanar, this can be efficiently tested.) If the spine ordering is already given, the problem of finding an edge partition into crossing-free pages is equivalent to that of properly -coloring circle graphs and hence determining the smallest such is NP-complete . While properly -coloring circle graphs is polynomial-time solvable for , it is open whether the problem becomes NP-hard for fixed . For the local page number we have NP-completeness for fixed spine ordering and each fixed .
For any , it is NP-complete to decide for a given graph and given spine ordering , whether there exists an edge partition such that is a -local book embedding.
2 Bounds in Terms of Density
Though not a fixed mathematical concept, the density of a graph quantifies the number of edges in terms of the number of vertices. An important specification of density is the maximum average degree of defined by
Recall that for a linear embedding of and a page we denote by the subgraph of on all edges in and all vertices of with at least one incident edge in . Clearly, if is crossing-free, then is outerplanar and thus . As and for each page , we immediately get an upper bound on the density of any graph with a -local book embedding.
For any graph we have
Let be any non-empty subgraph of a graph of local page number . Then there is a -local book embedding of , each page of which describes an outerplanar graph . Thus
In other words, the graph’s density gives a lower bound on all three kinds of page numbers. Perhaps surprisingly, there is also an upper bound on the union and local page numbers in terms of the graph’s density.
Nash-Williams  proved that any graph edge-partitions into forests if and only if
The smallest such , the arboricity of , thus satisfies . The star arboricity of is the minimum such that edge-partitions into star forests. Thus is the union -covering number of with respect to the class of all stars. Using the covering number framework, Knauer and the second author  introduced the corresponding local -covering number, the local star arboricity , as the minimum such that edge-partitions into some number of stars, but with each vertex having an incident edge in at most of these stars. It is known [1, 19] that and can be bound in terms of as
For any graph we have and . In particular, we have
Take an arbitrary spine ordering and an edge-partition into stars. Then each page is crossing-free, which shows . Now take an arbitrary spine ordering and an edge-partition into star forests. Then each connected component on each page is a star and thus crossing-free, which shows .
Though Theorem 2.1 is merely an observation, it has a number of interesting consequences. First of all, the local and union page number are not too far apart: . However, the local and union page numbers can be very far from the classical page number. For example, we have for every -regular graph , and hence and whenever is -regular. On the other hand, Malitz  proved that for every there are -vertex -regular graphs with page number . Together this proves Theorem 1.1.
For every planar graph we have and .
3 Planar Graphs
In this section we consider planar graphs. In particular, we prove Theorem 1.2 stating the existence of a planar graph with local page number . Our planar graph will be a large enough stacked triangulation (also known as planar -trees, chordal triangulations, or Apollonian networks). For this let and for define as obtained from by placing a new vertex in each facial triangle of , and connecting by edges to each of the three vertices of . Thus, for we have .
Suppose for the sake of contradiction there is a -local book embedding of . We consider the subgraphs of .
There exists an edge in with and .
Indeed, consider the four vertices of one of the two subgraphs in . Without loss of generality assume that . As for any , we can see as four pairwise incident edges in a multigraph on vertex set , where two vertices of are connected by an edge if there is common vertex of on the two respective pages. Thus, if were pairwise distinct, they would form a star, i.e., all pairwise intersections would be the same page . But then the whole subgraph on would be embedded on page , which is impossible as is not outerplanar.
So let be an edge in with . By the inductive construction of stacked triangulations, there is a set of seven vertices in that are incident to and and induce a path in ; see Fig. 2. By pigeon-hole principle and cyclic shifts of , we may assume that , where are consecutive in when restricted to . Each of and , , lies on or ; say . Then , and thus . In particular, we have .
Now observe that cannot be adjacent to any vertex with and . Indeed, such an edge would cross the edge and one of . Symmetrically, cannot be adjacent to any vertex with and . As induces a path in and no vertex of lies between and in , it follows that is an edge of the path. By symmetry assume . This implies that cannot be adjacent to any vertex with , as such an edge would cross the edges and .
But then form a facial triangle of with all three edges on page . However, there is no possible placement for the vertex in that is adjacent to each of . Thus, the planar graph admits no -local book embedding, which proves Theorem 1.2.
4 Graphs with Bounded Tree-Width
In this section we investigate the largest union page number and the largest local page number among all graphs of tree-width . Clearly it suffices to consider edge-maximal graphs of tree-width , the so-called -trees, which are inductively defined as follows: A graph is a -tree if and only if or is obtained from a smaller -tree by adding one new vertex whose neighborhood in is a clique of order .
As our main tool in this section, let us define a linear embedding to be a forest embedding if the edges on each page form a forest. For a graph , we say that a book embedding of some other graph contains a forest embedding of if there exists a set such that and restricted to is a forest embedding of .
For every and every -tree there exists a -tree such that every -local book embedding of contains a forest embedding of .
We find based on by induction on as follows.
In the base case we have and we find by induction on . In the base case of this inner induction we have and it suffices (for any ) to take . For , we get from induction a -tree all of whose -local book embeddings contain a forest embedding of . Starting with , add for each -clique in an independent set of vertices, together with all possible edges between and . The resulting graph has tree-width and hence can be augmented to a -tree . Consider any book embedding of . The inherited book embedding of contains a forest embedding of , i.e., we have a forest embedding of some -clique in . If one vertex in has its incident edges on pairwise different pages, then we have a forest embedding of , as desired. Otherwise, each vertex in has two incident edges on the same page in joining with two vertices in . By pigeon-hole principle, for a set of at least vertices of these are the same two vertices of . Since each of has incident edges on at most pages, again by pigeon-hole principle, one page in contains the edges between and at least vertices in . However this is a contradiction as is not outerplanar.
Now for the induction step of the outer induction, assume that is a -tree with vertices. Then is obtained from a -tree by adding one vertex whose neighborhood in is a clique of order . From induction we get a -tree all of whose -local book embeddings contain a forest embedding of . Now we can do the same argument as before: Obtain from by adding for each -clique in an independent set of size , together with all possible edges between and . Then any -local book embedding of induces an -local book embedding of , which hence contains a forest embedding of . Let be the -clique in that forms the neighborhood of in . The same argumentation as above then shows that at least one vertex in has its incident edges to on distinct pages, giving the desired forest embedding of . (Essentially, the only difference to the base case is that adding the independent sets to gives a full -tree, since is already a -tree.)
If admits a -local forest embedding , then
If is a -tree, then
To prove Theorem 1.3, we shall find for each a -tree whose local page number is at least . For there is nothing to show. For , let be any -tree with (Note that this is a vertex count!) and let be the corresponding -tree given by Lemma 2 for . Assuming for the sake of contradiction that , we obtain a -local forest embedding of . Then
a contradiction. Hence , as desired.
To end this section, let us also discuss some further implications of Lemma 2. We leave it open whether every -tree has local page number at most , i.e., whether the lower bound in Theorem 1.3 is tight. By Lemma 2 this is equivalent to every -tree admitting a -local forest book embedding. By putting each tree in each forest on a separate page, we even get a -local forest embedding with a tree on each page. Moreover, by Eq. 5 and Eq. 6 we would have , i.e., no more than trees in total, while at most at any one vertex.
And we get a similar statement for the maximum union page number of -trees. Suppose is an -union embedding of some graph, and that on all pages in together we have connected components. Putting each connected component on a separate (new) page, we obtain an -local book embedding with pages. Now if for all -trees, then Lemma 2 implies that every -tree even admits a -union forest embedding. Moreover, by Eq. 5 and Eq. 6 we get a forest embedding with trees in total, while having at most at any one vertex.
Specifically, in order to prove for every -tree , our task is to find a partition of the edges in into at most trees, such that every vertex is contained in no more than of these trees, as well as a spine ordering for which each of the trees is non-crossing. The first part has a very natural solution: Every -tree has chromatic number and admits a unique444Up to relabeling of color classes. proper -vertex coloring . Moreover, there are exactly pairs of colors in , any pair of color classes induces a tree in , and each vertex of is contained in exactly of these trees. Hence every -tree edge-partitions into trees with each vertex being contained in of these trees. Note that in this cover, every -clique in has all edges in pairwise distinct trees.
We have however not been able to prove (or disprove) the existence of a spine ordering under which no pair of color classes induces a crossing. If such exists, it would show for all and for odd and for even. Note that for the union page number we also need to group the trees into as few forests of vertex-disjoint trees as possible. Due to the nature of our coloring, this is equivalent to properly edge-coloring ; hence the distinction on the parity of .
5 Conclusions and Open Problems
In this paper we presented two novel graph parameters: the local page number and the union page number . Both parameters are weakenings of the classical page number and we have . Hence, one might be able to strengthen existing lower bounds of the form by showing or even . On the other hand, one might be able to support conjectured upper bounds of the form by showing the weaker bounds or even .
In this paper we started to pursue this direction of research. Let us list some concrete cases that are still open:
For the complete graph it is known  that . On the other hand, the density of implies that (Lemma 1). In Fig. 3 we indicate some -local book embeddings of for some small values of . According to this , , , and . Using the inequality from the proof of Lemma 1, we see that . (And with one further trick we get .) We refer to  for more details, and state it is an open problem to improve the following general bounds:
In 1989, Yannakakis  proved that for any planar graph we have , while removing an earlier claim  that there would be some planar graph with . Ganley and Heath  observed that stacked triangulation (using our notation from Section 3, but also known as the Goldner-Harary graph) is a planar graph with , which remains until today the best known lower bound. While , we show in Section 3 that , while we leave it as an open problem to improve on the bounds
We have a similar open problem for -trees, where we refer to the detailed discussion at the end of Section 4.
Besides determining the local and union page numbers for other graph classes (like for example regular graphs), it is also interesting to further analyze the relation between and . For example, what is the maximum of over all graphs ?
Finally, let us mention that changing the non-crossing condition Eq. 1 underlying the notion of book embeddings to for example a non-nesting condition, we get local and union versions of queue numbers. Interestingly, the proof of Theorem 2.1 remains valid and so does Corollary 1, giving that every planar graph has local queue number at most and union queue number at most .
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