Circle-Representations of Simple 4-Regular Planar Graphs

  • Michael A. Bekos
  • Chrysanthi N. Raftopoulou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


Lovász conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, (a) we affirmatively answer Lovász’s conjecture, if G is 3-connected, and, (b) we demonstrate an infinite class of connected 4-regular planar graphs which are not 3-connected and do not admit a realization as a system of circles.


Planar Graph Triangular Face Connected Planar Graph Contact Graph Tangent Circle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Biedl, T., Kant, G.: A Better Heuristic for Orthogonal Graph Drawings. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 24–35. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  2. 2.
    Bollobas, B.: Modern Graph Theory. Springer (1998)Google Scholar
  3. 3.
    Broersma, H., Duijvestijn, A., Göbel, F.: Generating all 3-connected 4-regular planar graphs from the octahedron graph. J. of Graph Theory 17(5), 613–620 (1993)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chernobelskiy, R., Cunningham, K., Goodrich, M., Kobourov, S.G., Trott, L.: Force-Directed Lombardi-Style Graph Drawing. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 310–321. Springer, Heidelberg (2012)Google Scholar
  5. 5.
    Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Drawing Trees with Perfect Angular Resolution and Polynomial Area. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 183–194. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Lombardi Drawings of Graphs. J. of Graph Algorithms and Applications 16(1), 85–108 (2011)CrossRefGoogle Scholar
  7. 7.
    Erdos, P., Renyi, A., Sos, V.T.: Combinatorial theory and its applications. North-Holland, Amsterdam (1970)Google Scholar
  8. 8.
    Hlinený, P.: Classes and recognition of curve contact graphs. J. of Combinatorial Theory, Series B 74(1), 87–103 (1998)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hlinený, P., Kratochvíl, J.: Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Mathematics 229(1-3), 101–124 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physikalische Klasse 88, 141–164 (1936)Google Scholar
  11. 11.
    Lehel, J.: Generating all 4-regular planar graphs from the graph of the octahedron. J. of Graph Theory 5(4), 423–426 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Leighton, F.T.: New lower bound techniques for VLSI. In: 22nd Ann. IEEE Symp. on Foundations of Computer Science, pp. 1–12. IEEE (1981)Google Scholar
  13. 13.
    Leiserson, C.E.: Area-efficient graph layouts (for VLSI). In: 21st Ann. IEEE Symp. on Foundations of Computer Science, vol. 1547, pp. 270–281. IEEE (1980)Google Scholar
  14. 14.
    Manca, P.: Generating all planar graphs regular of degree four. J. of Graph Theory 3(4), 357–364 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. of Computing 16, 421–444 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Tamassia, R., Tollis, I.: Planar grid embedding in linear time. IEEE Transactions on Circuits and Systems (1989)Google Scholar
  17. 17.
    Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Transaction on Computers 30(2), 135–140 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    West, D.B.: Introduction to Graph Theory. Prentice Hall (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael A. Bekos
    • 1
  • Chrysanthi N. Raftopoulou
    • 2
  1. 1.Institute for InformaticsUniversity of TübingenGermany
  2. 2.School of Applied Mathematical & Physical SciencesNational Technical University of AthensGreece

Personalised recommendations