NP membership of the chromatic coloring problem

Determine whether the chromatic coloring problem—computing the chromatic number χ(G) of a given graph G—belongs to the complexity class NP.

Background

The paper distinguishes the standard graph coloring decision problem from the optimization problem of computing the chromatic number, referring to the latter as chromatic coloring. While the decision version is well-studied, the authors explicitly remark that the optimization version is NP-hard and not known to be in NP.

Clarifying whether the optimization task of computing χ(G) admits polynomial-time verifiable certificates would settle its NP membership and refine our understanding of its complexity status beyond NP-hardness.