Abstract
We show that any k-th closed sphere-of-influence graph in a d-dimensional normed space has a vertex of degree less than \(5^d k\), thus obtaining a common generalization of results of Füredi and Loeb (Proc Am Math Soc 121(4):1063–1073, 1994 [1]) and Guibas et al. (Sphere-of-influence graphs in higher dimensions, Intuitive geometry [Szeged, 1991], 1994, pp. 131–137 [2]).
Dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays
Márton Naszódi acknowledges the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and the Hung. Nat. Sci. Found. (OTKA) grant PD104744. Part of this paper was written when Swanepoel visited EPFL in April 2015. Research by János Pach was supported in part by Swiss National Science Foundation grants 200020-144531 and 200020-162884.
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We thank the referee for helpful suggestions that improved the paper.
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Naszódi, M., Pach, J., Swanepoel, K. (2018). Sphere-of-Influence Graphs in Normed Spaces. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_16
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