Abstract
In a dispersable book embedding, the vertices of a graph G are ordered along a line \(\ell \), called spine, and the edges of G are drawn at different half-planes bounded by \(\ell \), called pages, such that: (i) no two edges of the same page cross, and (ii) no two edges of the same page share a common endvertex. The minimum number of pages needed in a dispersable book embedding of G is called its dispersable book thickness, dbt(G). Graph G is called dispersable if \(dbt(G)\) equals the maximum degree of G, \(\varDelta (G)\) (note that \(dbt(G) \ge \varDelta (G)\) always holds).
Back in 1979, Bernhart and Kainen conjectured that every k-regular bipartite graph G is dispersable and showed that it holds for \(k \in \{1, 2\}\). In this paper, we disprove the conjecture for the cases \(k=3\) (with a computer-aided proof), and \(k=4\) (with a purely combinatorial proof). In particular, we show that the bipartite 3-regular Gray graph has dispersable book thickness four, while the bipartite 4-regular Folkman graph has dispersable book thickness five. On the positive side, we show that every 3-connected 3-regular bipartite planar graph is dispersable.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
\([\mathsf {{A_1}},\mathsf {{A_2}}]\) is the union of \([\mathsf {{A_1}},\mathsf {{B_1}}]\), \([\mathsf {{B_1}},\mathsf {{C_1}}]\), \([\mathsf {{C_1}},\mathsf {{C_2}}]\), \([\mathsf {{C_2}},\mathsf {{A_2}}]\). As in the last three there are \(\mathsf {{C}}\)’s connectors including \(\mathsf {ac}\in [\mathsf {{C_1}},\mathsf {{C_2}}]\), the second \(\mathsf {{A}}\)’s connector can be only in \([\mathsf {{A_1}},\mathsf {{B_1}}]\).
References
Alam, J.M., Bekos, M.A., Gronemann, M., Kaufmann, M., Pupyrev, S.: On dispersable book embeddings. CoRR abs/1803.10030 (2018)
Barnette, D.W.: Conjecture 5. In: Tutte, W.T. (ed.) Recent Progress in Combinatorics, Proceedings of the Third Waterloo Conference on Combinatorics, pp. xiv+347. Academic Press, New York, London (1969)
Bekos, M.A., Gronemann, M., Raftopoulou, C.N.: Two-page book embeddings of 4-planar graphs. Algorithmica 75(1), 158–185 (2016)
Bekos, M.A., Kaufmann, M., Zielke, C.: The book embedding problem from a SAT-Solving perspective. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 125–138. Springer, Cham (2015)
Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Comb. Theory, Ser. B 27(3), 320–331 (1979)
Blankenship, R.: Book embeddings of graphs. Ph.D. thesis, Louisiana State University (2003)
Bouwer, I.: On edge but not vertex transitive regular graphs. J. Comb. Theory, Ser. B 12(1), 32–40 (1972)
Chartrand, G., Geller, D.P.: On uniquely colorable planar graphs. J. Comb. Theory 6(3), 271–278 (1969)
Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J. Algebraic Discret. Methods 8(1), 33–58 (1987)
Cornuéjols, G., Naddef, D., Pulleyblank, W.: Halin graphs and the travelling salesman problem. Math. Program. 26(3), 287–294 (1983)
Dujmović, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discret. Comput. Geom. 37(4), 641–670 (2007)
Dujmović, V., Wood, D.R.: On linear layouts of graphs. Discret. Math. Theor. Comput. Sci. 6(2), 339–358 (2004)
Folkman, J.: Regular line-symmetric graphs. J. Comb. Theory 3(3), 215–232 (1967)
de Fraysseix, H., de Mendez, P.O., Pach, J.: A left-first search algorithm for planar graphs. Discret. Comput. Geom. 13, 459–468 (1995)
Gerbracht, E.: Eleven unit distance embeddings of the Heawood graph. CoRR abs/0912.5395 (2009)
Heath, L.S.: Embedding planar graphs in seven pages. In: FOCS, pp. 74–83. IEEE Computer Society (1984)
Hoske, D.: Book embedding with fixed page assignments. Bachelor thesis, Karlsruhe Institute for Technology (2012)
Kainen, P.C.: Crossing-free matchings in regular outerplane drawings. In: Knots in Washington XXIX. George Washington Univ., Washington, DC, USA (2009). http://faculty.georgetown.edu/kainen/circLayouts.pdf
Kainen, P.C., Overbay, S.: Extension of a theorem of Whitney. Appl. Math. Lett. 20(7), 835–837 (2007)
Malitz, S.: Genus \(g\) graphs have pagenumber \(O(\sqrt{g})\). J. Algorithms 17(1), 85–109 (1994)
Malitz, S.: Graphs with E edges have pagenumber \(O(\sqrt{E})\). J. Algorithms 17(1), 71–84 (1994)
Nishizeki, T., Chiba, N.: Planar Graphs: Theory and Algorithms. Elsevier, New York (1988)
Overbay, S.B.: Generalized book embeddings. Ph.D. thesis, Colorado State University (1998)
Steinberg, R.: The state of the three color problem. In: Gimbel, J., Kennedy, J.W., Quintas, L.V. (eds.) Quo Vadis, Graph Theory?. Elsevier, New York (1993). Ann. Discret. Math. 55, 211–248
Wigderson, A.: The complexity of the Hamiltonian circuit problem for maximal planar graphs. Technical report TR-298, EECS Department, Princeton University (1982)
Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)
Acknowledgment
Our work is partially supported by DFG grant KA812/18-1. We would like to thank Prof. Paul Kainen for bringing this problem to our attention. We also thank Jessica Wolz for discussions on experimental aspects.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Alam, J.M., Bekos, M.A., Gronemann, M., Kaufmann, M., Pupyrev, S. (2018). On Dispersable Book Embeddings. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-00256-5_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-00255-8
Online ISBN: 978-3-030-00256-5
eBook Packages: Computer ScienceComputer Science (R0)