Abstract
A necessary and sufficient graph-theoretic condition is given for the number of different colorings, or regular configurations, on lattice points 1,2, … n in R to grow exponentially in n. This condition also characterizes when the largest eigenvalue of a zero-one matrix is greater than one. A similar but different condition is obtained for the coloring problem on the lattice points in R d , d ⩾ 2, with the hypercubic lattice structure.
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