On upper bounds for the multi-fold chromatic numbers of the plane

arXiv2206.12630MaRDI QIDQ6403129

Jaan Parts

Publication date: 25 June 2022

Abstract: The multi-fold chromatic number of the plane $ch{i}_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two corresponding $m$-subsets of $k$-set do not contain any common color. We consider upper bounds for $m$-fold chromatic numbers of the plane. Our main result is that for any $m$ the inequality $ch{i}_m<(1+2/sqrt3{)}^{2}cdotm+3.501$ holds.

Mathematics Subject Classification ID

Planar graphs; geometric and topological aspects of graph theory (05C10) Coloring of graphs and hypergraphs (05C15)

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