## Drawing Subdivisions Of Complete And Complete Bipartite Graphs On Books (1999)

Citations: | 9 - 0 self |

### BibTeX

@TECHREPORT{Blankenship99drawingsubdivisions,

author = {Robin Blankenship and Bogdan Oporowski},

title = {Drawing Subdivisions Of Complete And Complete Bipartite Graphs On Books},

institution = {},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We investigate book-thickness of subdivided complete and subdivided complete bipartite graphs. We discuss well-known results that the book-thickness of each of Kn and Kn;n is large when n is large, while, for every n, some subdivision of Kn and some subdivision of Kn;n have bookthickness at most three. The main result of this paper, whose proof is based on Ramsey theory, states that every graph obtained from Kn and Kn;n by subdividing each edge at most once has large book-thickness when n is large. Some generalizations of this result are also discussed. 1. Introduction Graph theory is a very youthful and vibrant part of mathematics. Many of its problems and results are readily accessible to a general audience. One of its particularly attractive areas, topological graph theory, deals with embedding graphs, viewed as topological spaces, into other topological spaces. In this paper, we will focus on embedding particular kinds of graphs: subdivisions of complete graphs and subdivisions ...

### Citations

1334 |
Graph Theory with Applications
- Bondy, Murty
- 1976
(Show Context)
Citation Context ...mplete bipartite graphs, into a particular kind of topological space, namely, a book. We will consider graphs that are finite and have no loops and no multiple edges. For basic graph terminology, see =-=[2]-=-. A graph G can be embedded in a topological space X if the vertices of G can be represented by distinct elements of X and each edge of G can be represented by an arc in X, that is, the image of a 1--... |

173 |
Sur le problèmes des courbes gauches en topologie, Fund
- Kuratowski
(Show Context)
Citation Context ...Clearly, a graph can be embedded in a 1-page book if and only if it can be embedded in a 2-page book. The latter occurs exactly when the graph is planar. The following classical theorem of Kuratowski =-=[6]-=- gives an elegant characterization of such graphs. Theorem 1.1. A graph is planar if and only if none of its subgraphs is isomorphic to a subdivision of K 5 or of K 3;3 . The following is a surprising... |

52 | Embedding graphs in books: a layout problem with applications to VLSI design
- Chung, Leighton, et al.
- 1987
(Show Context)
Citation Context ...arly possible since the union of pages 2 and 3 forms a plane. The last theorem means that arbitrary book embeddings are somewhat uninteresting. Motivated by such practical applications as VLSI design =-=[3]-=-, we now focus on a particular type of book embedding. Specifically, we require that every vertex of G be embedded on the spine, and that the interior of every edge lie on a single page. The least num... |

27 |
Genus g graphs have pagenumber O( √ g
- Malitz
- 1994
(Show Context)
Citation Context ...graph whose book-thickness exceeds two. Endo [4] proved the following: Theorem 2.5. Every graph that embeds in a torus has book-thickness at most seven. For similar, more general results, see [5] and =-=[7]-=-. 4 BLANKENSHIP AND OPOROWSKI 3. Main Result This section contains the statement and proof of the main result of the paper. Roughly speaking, this result states that, for large n, the graphs obtained ... |

16 |
The pagenumber of genus g graphs is O(g
- HEATH, ISTRAIL
- 1992
(Show Context)
Citation Context ... planar graph whose book-thickness exceeds two. Endo [4] proved the following: Theorem 2.5. Every graph that embeds in a torus has book-thickness at most seven. For similar, more general results, see =-=[5]-=- and [7]. 4 BLANKENSHIP AND OPOROWSKI 3. Main Result This section contains the statement and proof of the main result of the paper. Roughly speaking, this result states that, for large n, the graphs o... |

16 |
Four pages are necessary and sufficient for planar graphs
- Yannakakis
- 1986
(Show Context)
Citation Context ...ossible, for example, BT(K 3;3 ) = 3 6= d 3 2 e. The book-thickness of a graph can be bounded from above in terms of the simplest surface in which the graph can be embedded. In particular, Yannakakis =-=[9]-=- proved the following: Theorem 2.4. Every planar graph has book-thickness at most four. The bound in Theorem 2.4 is best possible---there are planar graphs whose bookthickness is four. However, the co... |

10 |
On the embeddability of compacta in n-books: intrinsic and extrinsic properties
- ATNEOSEN
- 1968
(Show Context)
Citation Context ...space X if the vertices of G can be represented by distinct elements of X and each edge of G can be represented by an arc in X, that is, the image of a 1--1 continuous function from the unit interval =-=[0; 1]-=- into X. Moreover, the endpoints of every such arc correspond to the endvertices of the edge represented by the arc, and the interior of every such arc avoids all other arcs and all points representin... |

2 |
The pagenumber of toroidal graphs is at most seven. Discrete Mathematics
- Endo
- 1997
(Show Context)
Citation Context ...e planar graphs whose bookthickness is four. However, the construction of such graphs is quite difficult (see [9]). The reader is invited to find a planar graph whose book-thickness exceeds two. Endo =-=[4]-=- proved the following: Theorem 2.5. Every graph that embeds in a torus has book-thickness at most seven. For similar, more general results, see [5] and [7]. 4 BLANKENSHIP AND OPOROWSKI 3. Main Result ... |