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The splitting number of the 4-cube

  • Luerbio Faria
  • Celina Miraglia Herrera de Figueiredo
  • Candido Ferreira Xavier de MendonÇa Neto
Conference paper
Part of the Lecture Notes in Computer Science book series (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 v1 and v2, and attaches the neighbors of v either to v1 or to v2. 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).

Keywords

Complete Bipartite Graph Black Vertex Layout Algorithm Auxiliary Graph Nonadjacent Vertex 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Luerbio Faria
    • 1
    • 3
  • Celina Miraglia Herrera de Figueiredo
    • 2
    • 3
  • Candido Ferreira Xavier de MendonÇa Neto
    • 4
  1. 1.Faculdade de FormaÇÃo de ProfessoresUERJBrazil
  2. 2.Instituto de MatemáticaUFRJBrazil
  3. 3.COPPE Sistemas e ComputaÇÃoUFRJBrazil
  4. 4.Instituto de ComputaÇÃoUNICAMPBrazil

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