Exact packing chromatic numbers of the three distance graphs

Determine the exact packing chromatic numbers of the distance graphs D(1,6), D(1,8), and D(1,9), whose currently established bounds are 17≤χρ(D(1,6))≤20, 18≤χρ(D(1,8))≤22, and χρ(D(1,9))≤17.

Background

The paper studies packing colorings of integer distance graphs D(1,t), where vertices are the integers and adjacency is defined by differences of 1 or t. For D(1,6), the paper establishes density-based lower and upper bounds yielding 17≤χρ(D(1,6))≤20; for D(1,8), it obtains 18≤χρ(D(1,8))≤22; and for D(1,9), it provides the upper bound χρ(D(1,9))≤17.

The density certificates, periodic constructions, and supplementary exclusion arguments do not close these gaps. Consequently, the precise minimum number of colors required for packing colorings of these three distance graphs remains unresolved.