Chromaticity of Complete Tripartite Graphs With Certain Star or Matching Deleted

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 6-partite graphs with 6n vertices according to the number of 7-independent partitions of G. Using these results, we investigate the chromaticity of G with certain star or matching deleted. As a byproduct, many new families of chromatically unique complete 6-partite graphs with certain star or matching deleted are obtained. 2000 Mathematical Subject Classification. Primary 05C15.