On chromatic uniqueness of certain 5-partite graphs

Abstract

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 with 5n+3 vertices according to the number of 6-independent partitions of G. Using these results, we investigate the chromaticity of G with certain star or matching deleted. As a by-product, many new families of chromatically unique complete 5-partite graphs with certain star or matching deleted are obtained.