A graph $G$ has a $k$-page book embedding if $G$ can be embedded into a $k$-page book. The minimum $k$ such that $G$ has a $k$-page book embedding is the book thickness of $G$, denoted $bt(G)$. Most of the work on this subject has been done for unoriented graphs and oriented acyclic graphs (no directed cycles). In this work we discuss oriented graphs $\overrightarrow{D}$ containing directed cycles by using oriented book embeddings and oriented book thickness, $obt(\overrightarrow{D})$. To characterize $\overrightarrow{D}$ such that $obt(\overrightarrow{D}) = k$, we define the class $\mathcal{M}^k$ of $k$-page critical oriented graphs to be all oriented graphs $\overrightarrow{D}$ with $obt(\overrightarrow{D}) =k$, but for every proper oriented subgraph of $\overrightarrow{D}$, denoted $\overrightarrow{D}'$, we have that $obt(\overrightarrow{D}') < k$. Determining $\mathcal{M}^k$ for general $k$ is challenging; we narrow down the list of oriented graphs in $\mathcal{M}^k$ for small $k$. In this work we show complete lists for $\mathcal{M}^1$ and for $\mathcal{M}^2 \cap \mathcal{U}$, where $\mathcal{U}$ consists of all strictly dicyclic oriented graphs, that is, oriented graphs containing exactly one oriented cycle, which is a directed cycle.
Keywords: book embedding, book thickness, oriented book embedding, oriented book thickness, directed cycle, critical graph
The $n$-$book ~embedding$ of a graph $G$ is an embedding of the graph $G$ in an $n$-book with the vertices of $G$ on the spine and each edge to the pages without crossing each other. If the degree of vertices of $G$ at most one in each page, the $n$-book embedding is $matching$. The $matching~ book~ thickness$ of graph $G$ is the smallest $n$ so that $G$ can be matching embedded in an $n$-book. In this paper, the matching book thickness of the biconnected outerplanar graphs is determined.
It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. The latter extends a theorem of H. Whitney and gives two-page book embeddings for $X$-trees and square grids.
A book embedding of a graph consists of an embedding of its vertices along the spine of a book, and an embedding of its edges on the pages such that edges embedded on the same page do not intersect. The pagenumber is the minimum number of pages in which the graph $G$ can be embedded. The main purpose of this paper is to study the book embedding of the complete expansion graph. This is the first work about it, and some exact pagenumbers of the complete expansion graphs of some special graphs are obtained.
The $book$ $embedding$ of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. The book embedding is $matching$ if the pages have maximum degree $1$. The $matching$ $book$ $thickness$ is the minimum number of pages in which graphs can be matching book embedded. In this paper, we show that the matching book thickness of the Cartesian product $K_p\Box C_q$of a complete graph $K_p$ and a cycle $C_q$ is equal to $Δ(K_p\Box C_q)+1$.
A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded is called the \emph{page number}. The book-embedding of graphs may be important in several technical applications, e.g., sorting with parallel stacks, fault-tolerant processor arrays design, and layout problems with application to very large scale integration (VLSI). Bernhart and Kainen firstly considered the book-embedding of the planar graph and conjectur…
In a dispersable book embedding, the vertices of a given graph $G$ must be ordered along a line l, called spine, and the edges of G must be drawn at different half-planes bounded by l, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graphs induced by the edges of each page are 1-regular. The minimum number of pages needed by any dispersable book embedding of $G$ is referred to as the dispersable book thickness $dbt(G)$ of $G$. Graph $G$ is called dispersable if $dbt(G) = Δ(G)$ holds (note that $Δ(G) \leq dbt(G)$ always holds). Back in 1979, Bernhart a…
The book embedding of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. A book embedding is matching if the vertices in the same page have maximum degree at most 1. The matching book thickness is the minimum number of pages in which a graph can be matching book embedded. A graph $G$ is dispersable if and only if $mbt(G)=Δ(G)$. In this paper, we prove that bipartite cubic planar graphs are dispersable.
The \emph{matching book embedding} of a graph $G$ is to arrange its vertices on the spine, and draw its edges into the pages so that the edges on every page do not intersect each other and the maximum degree of vertices on every page is one. The \emph{matching book thickness} is the minimum number of pages in which the graph $G$ can be matching embedded. In this paper, the matching book thickness of Halin graphs is determined.
A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of g…
We conjecture that every oriented graph $G$ on $n$ vertices with $δ^+ (G) , δ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered.
Back in the Eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book-embedding viewpoint of the proble…
An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this pro…
In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges on the same page do not cross. In this paper, we prove that every $1$-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is $O(\sqrt{n})$, where $n$ is the number of vertices of the graph.
A $k$-stack layout (also called a $k$-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing edges with respect to the vertex order. The stack number (book thickness, page number) of a graph is the minimum $k$ such that it admits a $k$-stack layout. A $k$-queue layout is defined similarly, except that no two edges in a single set may be nested. It was recently proved that graphs of various non-minor-closed classes are subgraphs of the strong product of a path and a graph with bounded treewidth. Motivated by this dec…
Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed cycle of length l. We prove this conjecture asymptotically for the case when l is large enough compared to k and k>6. The case when k<7 was already settled asymptotically by Kelly, Kuehn and Osthus.
In this work we investigate the complexity of some problems related to the {\em Simultaneous Embedding with Fixed Edges} (SEFE) of $k$ planar graphs and the PARTITIONED $k$-PAGE BOOK EMBEDDING (PBE-$k$) problems, which are known to be equivalent under certain conditions. While the computational complexity of SEFE for $k=2$ is still a central open question in Graph Drawing, the problem is NP-complete for $k \geq 3$ [Gassner {\em et al.}, WG '06], even if the intersection graph is the same for each pair of graphs ({\em sunflower intersection}) [Schaefer, JGAA (2013)]. We improve on these results…
We use a randomised embedding method to prove that for all α>0 any sufficiently large oriented graph G with minimum in-degree and out-degree δ^+(G),δ^-(G)\geq (3/8+α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.
A "book" with k pages consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The pagenumber of a graph G is the minimum k such that G admits a k-page embedding (that is, a k-page drawing with no edge crossings). The k-page crossing number nu_k(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the p…
We find the minimal number of links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete $k$-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.
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…
