Packing domatic chromatic numbers for larger infinite-path layers

Determine χ_{ρ,k}(P_∞) for every integer k≥7.

Background

The paper determines the exact values of χ{ρ,k}(P∞) for k∈{3,4,5}, bounds the value for k=6, and proves the general lower bound χ{ρ,k}(P∞)>(k−1)/2. No exact values are established for k≥7, which the authors explicitly identify as unresolved.

References

The cases $k\geq 7$ remain open and are left for future investigation.

— Partitioning an $S$-packing coloring into broadcast dominating sets  (2610.03477 - Brešar et al., 2 Oct 2026) in Section 3.2, The infinite path