Exact value for six dominating broadcasts on the infinite path

Determine whether χ_{ρ,6}(P_∞)=4 or χ_{ρ,6}(P_∞)=5 for the two-way infinite path.

Background

Theorem 3.4 proves that χ{ρ,6}(P∞) lies in {4,5} by giving both a lower bound of 4 and a construction using 5 colors. The paper does not determine which of these two values is attained. The subsequent cases k≥7 are also explicitly left unresolved.

References

For $k=6$, the problem is reduced to determining which of the two possible values in ${4,5}$ is attained by $\chi_{\rho,6}(P_\infty)$, a question we leave for the reader.

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