Abstract
The invisibility graph I(X) of a set \(X \subseteq {\mathbb R}^d\) is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X.
We settle a conjecture of Matoušek and Valtr claiming that for invisibility graphs of planar sets, the chromatic number cannot be bounded in terms of the clique number.
Work on this paper was supported by the project 1M0545 of the Ministry of Education of the Czech Republic. The third author was also supported by OTKA Grant K76099 and by the grant no. MSM0021620838 of the Ministry of Education of the Czech Republic. The first and fourth authors were also supported by the Czech Science Foundation under the contract no. 201/09/H057. The first, second and fifth authors were partially supported by project GAUK 52110.
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References
Breen, M., Kay, D.C.: General decomposition theorems for m-convex sets in the plane. Israel Journal of Mathematics 24, 217–233 (1976)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)
Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)
Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)
Matoušek, J., Valtr, P.: On visibility and covering by convex sets. Israel Journal of Mathematics 113, 341–379 (1999)
Perles, M., Shelah, S.: A closed (n + 1)-convex set in ℝ2 is a union of n 6 convex sets. Israel Journal of Mathematics 70, 305–312 (1990)
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Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R., Valtr, P. (2010). On Three Parameters of Invisibility Graphs. In: Thai, M.T., Sahni, S. (eds) Computing and Combinatorics. COCOON 2010. Lecture Notes in Computer Science, vol 6196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14031-0_22
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DOI: https://doi.org/10.1007/978-3-642-14031-0_22
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