Optimal tristance anticodes in certain graphs
Journal of Combinatorial Theory Series A
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For $z_1,z_2,z_3\in\Z^2$, the tristance $d_3(z_1,z_2,z_3)$ is a generalization of the $L_1$-distance on $\mathbb{Z}^2$ to a quality that reflects the relative dispersion of three points rather than two. In this paper we prove that at least 3k2 colors are required to color the points of $\mathbb{Z}^2$, such that the tristance between any three distinct points, colored with the same color, is at least 4k. We prove that 3k2+3k+1 colors are required if the tristance is at least 4k+2. For the first case we show an infinite family of colorings with colors and conjecture that these are the only colorings with 3k2 colors.