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New upper bound for the chromatic number of a random subgraph of a distance graph

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Abstract

This paper is related to the classical Hadwiger-Nelson problem dealing with the chromatic numbers of distance graphs in ℝn. We consider the class consisting of the graphs G(n, 2s + 1, s) = (V (n, 2s + 1), E(n, 2s + 1, s)) defined as follows:

$V(n,2s + 1) = \{ x = (x_1 ,x_2 ,...,x_n ):x_i \in \{ 0,1\} ,x_1 + x_2 + \cdots + x_n = 2s + 1\} ,E(n,2s + 1,s) = \{ \{ x,y\} :(x,y) = s\} ,$

where (x, y) stands for the inner product. We study the random graph G(G(n, 2s + 1, s), p) each of whose edges is taken from the set E(n, 2s+1, s) with probability p independently of the other edges. We prove a new bound for the chromatic number of such a graph.

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  1. Lomonosov Moscow State University, Moscow, Russia

    A. S. Gusev

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  1. A. S. Gusev
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Gusev, A.S. New upper bound for the chromatic number of a random subgraph of a distance graph. Math Notes 97, 326–332 (2015). https://doi.org/10.1134/S0001434615030037

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