Chromaticity of complete 4-partite graphs with certain star or matching deleted

Abstract

Let P(G,?) 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,?) = P(H,?). 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 4-partite graphs G accordingly to the number of 5-independent partitions of G. Using these results, we investigate the chromaticity of G with certain star or matching deleted. As a by-product, we obtain new families of chromatically unique complete 4-partite graphs with certain star or matching deleted.