Chromaticity of Complete 5-Partite Graphs with Certain Star or Matching Deleted

Ameen Shaman Ameen, Yee Hock Peng, Haixing Zhao, Gee Choon Lau, Roslan Hasni

Abstract

Let P(G, \lambda) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted by G ∼ H, if P(G,\lambda ) = P(H,\lambda ). 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 + 1 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.