Interpolation theorem for partial Grundy coloring

An ordered partition Π = ( V 1 , V 2 , … , V k ) of G is a partial Grundy coloring if for each i , 2 ≤ i ≤ k , there exists a vertex x i in V i such that x i is adjacent to at least one vertex in V j for each j < i . The partial Grundy number of G is the largest positive integer k for which G has a partial Grundy coloring using k colors and it is denoted by ∂ Γ ( G ) . For any graph G , we have χ ( G ) ≤ ∂ Γ ( G ) . P. Erdős, S.T. Hedetniemi, R.C. Laskar and G.C.E. Prins (2003) [2] raised the following question: For any graph G and any positive integer k with χ ( G ) ≤ k ≤ ∂ Γ ( G ) , does G have a partial Grundy coloring using k colors? In this note, we provide an affirmative answer to this question.