Chromaticity of a family of 5-partite graphs

Let P(G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G,λ) = P(H,λ). We write [G] = {H∣H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n vertices according to the number of 6-independent partitions of G. Using these results, we investigate the chromaticity of G with certain stars or matching deleted parts . As a by-product, two new families of chromatically unique complete 5-partite graphs G with certain stars or matching deleted parts are obtained.