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Universal Sets for Straight-Line Embeddings of Bicolored Graphs

In: Thirty Essays on Geometric Graph Theory

Author

Listed:
  • Josef Cibulka

    (Charles University, Department of Applied Mathematics)

  • Jan Kynčl

    (Charles University, Department of Applied Mathematics and Institute for Theoretical Computer Science)

  • Viola Mészáros

    (Charles University, Department of Applied Mathematics and Institute for Theoretical Computer Science
    Technical University of Berlin, Institute of Mathematics)

  • Rudolf Stolař

    (Charles University, Department of Applied Mathematics)

  • Pavel Valtr

    (Charles University, Department of Applied Mathematics and Institute for Theoretical Computer Science)

Abstract

A set S of n points is 2-color universal for a graph G on n vertices if, for every proper 2-coloring of G and for every 2-coloring of S with the same sizes of color classes as G, G is straight-line embeddable on S. We show that the so-called double-chain is 2-color universal for paths if each of the two chains contains at least one fifth of all the points, but not if one of the chains is more than approximately 28 times longer than the other. A 2-coloring of G is equitable if the sizes of the color classes differ by at most 1. A bipartite graph is equitable if it admits an equitable proper coloring. We study the case when S is the double-chain with chain sizes differing by at most 1 and G is an equitable bipartite graph. We prove that this S is not 2-color universal if G is not a forest of caterpillars and that it is 2-color universal for equitable caterpillars with at most one half nonleaf vertices. We also show that if this S is equitably 2-colored, then equitably properly 2-colored forests of stars can be embedded on it.

Suggested Citation

  • Josef Cibulka & Jan Kynčl & Viola Mészáros & Rudolf Stolař & Pavel Valtr, 2013. "Universal Sets for Straight-Line Embeddings of Bicolored Graphs," Springer Books, in: János Pach (ed.), Thirty Essays on Geometric Graph Theory, pages 101-119, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-0110-0_8
    DOI: 10.1007/978-1-4614-0110-0_8
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