Abstract
It is proved that each distance graph on a plane has an induced subgraph with a chromatic number that is at most 4 containing over 91% of the vertices of the original graph. This result is used to obtain the asymptotic growth rate for a threshold probability that a random graph is isomorphic to a certain distance graph on a plane. Several generalizations to larger dimensions are proposed.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 51, Topology, 2013.
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Kokotkin, A., Raigorodskii, A. On Large Subgraphs with Small Chromatic Numbers Contained in Distance Graphs. J Math Sci 214, 665–674 (2016). https://doi.org/10.1007/s10958-016-2804-3
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DOI: https://doi.org/10.1007/s10958-016-2804-3