Interpolation theorem for partial Grundy coloring

Abstract

An ordered partition Π=(V1,V2,…,Vk) of G is a partial Grundy coloring if for each i, 2≤i≤k, there exists a vertex xi in Vi such that xi is adjacent to at least one vertex in Vj 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.