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.
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Acknowledgements
We thank Jakub Černý for his active participation in the earlier stages of our discussions.
Research was supported by the Center of Excellence – Institute for Theoretical Computer Science, Prague (project P202/12/G061 of GA CR) and by Grant SVV-2012-265313 (Discrete Models and Algorithms). Viola Mészáros was also partially supported by ESF EuroGiga project ComPose (IP03), by OTKA Grant K76099 and by OTKA Grant 102029. Josef Cibulka, Jan Kynčl, and Pavel Valtr were also supported by the Grant Agency of Charles University, GAUK 52410.
Part of the research was conducted during the Special Semester on Discrete and Computational Geometry at École Polytechnique Fédérale de Lausanne, organized and supported by the CIB (Centre Interfacultaire Bernoulli) and the SNSF (Swiss National Science Foundation).
A preliminary version appeared in proceedings of Graph Drawing 2008 [7].
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Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R., Valtr, P. (2013). Universal Sets for Straight-Line Embeddings of Bicolored Graphs. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_8
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