Chromaticity of certain tripartite graphs identified with a path

For a graphG, let P(G) be its chromatic polynomial. Two graphsG andH are chromatically equivalent if P(G)=P(H). A graph G is chromatically unique if P(H)= P(G) implies that H G. In this paper, we classify the chromatic classes of graphs obtained from K2,2,2 ∪ Pm(m 3), (K2,2,2 − e) ∪ Pm(m 5) and (K2,2,2 − 2e) ∪ Pm(m 6) by identifying the end-vertices of the path Pm with any two vertices of K2,2,2, K2,2,2 − e and K2,2,2 − 2e, respectively, where e and 2e are, respectively, an edge and any two edges ofK2,2,2. As a by-product of this, we obtain some families of chromatically unique and chromatically equivalent classes of graphs. © 2006 Elsevier B.V. All rights reserved.