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Subdivision Number of Large Complete Graphs and Large Complete Multipartite Graphs

  • Severino V. Gervacio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)

Abstract

A graph whose vertices can be represented by distinct points in the plane such that points representing adjacent vertices are 1 unit apart is called a unit-distance graph. Not all graphs are unit distance graphs. However, if every edge of a graph is subdivided by inserting a new vertex, then the resulting graph is a unit-distance graph. The minimum number of new vertices to be inserted in the edges of a graph G to obtain a unit-distance graph is called the subdivision number of G, denoted by sd(G). We show here in a different and easier way the known result sd(K m,n ) = (m − 1)(n − m) when nm(m–1). This result is used to show that the subdivision number of the complete graph is asymptotic to (\(n \above 0pt 2\)), its number of edges. Likewise, the subdivision number of the complete bipartite graph K m, n is asymptotic to mn, its number of edges. More generally, the subdivision number of the complete n-partite graph is asymptotic to its number of edges.

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References

  1. 1.
    Gervacio, S.V., Maehara, H.: Subdividing a graph toward a unit distance graph in the plane. European Journal of Combinatorics 21, 223–229 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Severino V. Gervacio
    • 1
  1. 1.Department of MathematicsDe La Salle UniversityPhilippines

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