Additive subadditivity of closed chromatic numbers

Determine whether, for every graph G and integers k_1,k_2 with positive integer n, the existence of closed chromatic numbers χ_{n,k_1}(G) and χ_{n,k_2}(G) implies the additive bound χ_{n,k_1+k_2}(G) ≤ χ_{n,k_1}(G)+χ_{n,k_2}(G).

Background

Theorem 2.5 establishes the multiplicative estimate χ{n,k_1+k_2}(G) ≤ χ{n,k_1}(G)χ_{n,k_2}(G) whenever the relevant closed chromatic numbers exist. The authors observe that this bound appears rough and ask whether the sum of the two individual closed chromatic numbers always provides a sharper upper bound.

The problem concerns closed colorings with prescribed neighborhood-sum remainders modulo n. It asks whether two proper closed colorings can always be combined while using no more colors than the sum of their orders.

References

Our bound on $\chi_{n,k_1+k_2}(G)$ in the last statement of Theorem \ref{thm: k mod n units and divisor first relations} seems rather rough. In particular, it is natural to ask the following question: Let $k_1,k_2\inZ$ and $n\inZ+$, and let $\chi_{n,k_1}(G)$ and $\chi_{n,k_2}(G)$ exist. Does this imply $\chi_{n,k_1+k_2}(G) \leq \chi_{n,k_1}(G) +\chi_{n,k_2}(G)$?

— Chromatic numbers with closed local modular constraints  (2503.00406 - Herden et al., 1 Mar 2025) in Question following Theorem 2.5, Section 2 (Basic Results)