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Latin American Symposium on Theoretical Informatics

LATIN 1998: LATIN'98: Theoretical Informatics pp 141–150Cite as

The splitting number of the 4-cube

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1380))

Abstract

The splitting number of a graph G is the smallest integer k ≥ 0 such that a planar graph can be obtained from G by k splitting operations. Such operation replaces v by two nonadjacent vertices v 1 and v 2, and attaches the neighbors of v either to v 1 or to v 2. The n-cube has a distinguished plaice in Computer Science. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4-cube is 8, but no results about splitting number of other nonplanar n-cubes are known. In this note we give a proof that the splitting number of the 4-cube is 4. In addition, we give the lower bound 2n−2 for the splitting number of the n-cube. It is known that the splitting number of the n-cube is O(2n), thus our result implies that the splitting number of the n-cube is λ(2n).

Work partially supported by CNPq, CAPES, FAPERJ and FAPESP, Brazilian research agencies.

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References

  1. M. S. Anderson, R. B. Richter and P. Rodney (1996). “The crossing number of C 6×C 6”, Congressus Numerantium 118, 97–107.

    MathSciNet  Google Scholar 

  2. M. S. Anderson, R. B. Richter and P. Rodney (1997). “The crossing number of C 7×C 7”. Proc. 28th Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Florida, USA.

    Google Scholar 

  3. R. J. Cimikowski (1992). “Graph planarization and skewness”, Congressus Numerantium, 88, 21–32.

    MATH  MathSciNet  Google Scholar 

  4. A. M. Dean and R. B. Richter (1995). “The crossing number of C 4×C 4”, Journal of Graph Theory 19, 125–129.

    MathSciNet  Google Scholar 

  5. P. Eades and C. F. X. MendonÇa (1993). “Heuristics for Planarization by Vertex Splitting”. Proc. ALCOM Int. Workshop on Graph Drawing, GD'93, 83–85.

    Google Scholar 

  6. P. Eades and C. F. X. MendonÇa (1996). “Vertex Splitting and Tension-Free Layout”. Proc. GD'95, Lecture Notes in Computer Science 1027, 202–211.

    Google Scholar 

  7. R. B. Eggleton and R. P. Guy (1970). “The crossing number of the n-cube”, AMS Notices 17, 757.

    Google Scholar 

  8. L. Faria (1994). “Bounds for the crossing number of the n-cube”, Master thesis, Universidade Federal do Rio de Janeiro (In Portuguese).

    Google Scholar 

  9. L. Faria and C. M. H. Figueiredo (1997). “On the Eggleton and Guy conjectured upper bound for the crossing number of the n-cube”, submitted to Math. Slovaca.

    Google Scholar 

  10. L. Faria, C. M. H. Figueiredo and C. F. X. MendonÇa (1997). “Splitting number is NP-Complete”, Technical Report ES-443/97, COPPE/UFRJ, Brazil.

    Google Scholar 

  11. M. R. Garey and D. S. Johnson (1983). “Crossing number is NP-complete”, SIAM J. Algebraic and Discrete Methods 4, 312–316.

    MathSciNet  Google Scholar 

  12. F. Harary, P. C. Kainen and A. J. Schwenk (1973). “Toroidal graphs with arbitrarily high crossing number”, Nanta Math. 6, 58–67.

    MathSciNet  Google Scholar 

  13. N. Hartfield, B. Jackson and G. Ringel (1985). “The splitting number of the complete graph”, Graphs and Combinatorics 1, 311–329.

    MathSciNet  Google Scholar 

  14. B. Jackson and G. Ringel (1984). “The splitting number of complete bipartite graphs”, Arch. Math. 42, 178–184.

    Article  MathSciNet  Google Scholar 

  15. K. Kuratowski (1930). “Sur le problème des courbes gauches en topologie”, Fundamenta Mathematicae 15, 271–283.

    MATH  Google Scholar 

  16. F. T. Leighton (1981). “New lower bound techniques for VLSI”, Proc. 22nd Annual Symposium on Foundations of Computer Science, Long Beach CA, 1–12.

    Google Scholar 

  17. A. Liebers (1996). “Methods for Planarizing Graphs — A Survey and Annotated Bibliography”, ftp://ftp.informatik.uni-konstanz.de/pub/preprints/1996/preprint-012.ps.Z.

    Google Scholar 

  18. P. C. Liu and R. C. Geldmacher (1979). “On the deletion of nonplanar edges of a graph”, Congressus Numerantium 24, 727–738.

    MathSciNet  Google Scholar 

  19. T. Madej (1991). “Bounds for the crossing number of the n-cube”, Journal of Graph Theory 15, 81–97.

    MATH  MathSciNet  Google Scholar 

  20. C. F. X. MendonÇa (1994). “A Layout System for Information System Diagrams”, Ph.D. thesis, University of Queensland, Australia.

    Google Scholar 

  21. R. D. Ringeisen and L. W. Beineke (1978). “The crossing number of C 3×C n”, Journal of Combinatorial Theory Ser. B 24, 134–136.

    Article  MathSciNet  Google Scholar 

  22. O. Sýkora and I. Vrto (1993). “On the crossing number of hypercubes and cube connected cycles”, BIT 33, 232–237.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculdade de FormaÇÃo de Professores, UERJ, Brazil

    Luerbio Faria

  2. Instituto de Matemática, UFRJ, Brazil

    Celina Miraglia Herrera de Figueiredo

  3. COPPE Sistemas e ComputaÇÃo, UFRJ, Brazil

    Luerbio Faria & Celina Miraglia Herrera de Figueiredo

  4. Instituto de ComputaÇÃo, UNICAMP, Brazil

    Candido Ferreira Xavier de MendonÇa Neto

Authors
  1. Luerbio Faria
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  2. Celina Miraglia Herrera de Figueiredo
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  3. Candido Ferreira Xavier de MendonÇa Neto
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Editor information

Cláudio L. Lucchesi Arnaldo V. Moura

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© 1998 Springer-Verlag Berlin Heidelberg

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Faria, L., de Figueiredo, C.M.H., de MendonÇa Neto, C.F.X. (1998). The splitting number of the 4-cube. In: Lucchesi, C.L., Moura, A.V. (eds) LATIN'98: Theoretical Informatics. LATIN 1998. Lecture Notes in Computer Science, vol 1380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054317

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  • DOI: https://doi.org/10.1007/BFb0054317

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64275-6

  • Online ISBN: 978-3-540-69715-2

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