Chromaticity of certain tripartite graphs identified with a path (Q860454)

This is a rather technical paper, so the authors' abstract does the best job at summarizing it: ``For a graph \(G\), let \(P(G)\) be its chromatic polynomial. Two graphs \(G\) and \(H\) are chromatically equivalent if \(P(G)=P(H)\). A graph \(G\) is chromatically unique if \(P(H)=P(G)\) implies \(H \cong G\). In this paper we classify the chromatic classes of graphs obtained from \(K_{2,2,2}\cup P_m\) (\(m\geq 3\)), \((K_{2,2,2}-e)\cup P_m\) (\(m\geq 5\)) and \((K_{2,2,2}-2e)\cup P_m\) (\(m\geq 6\)) by identifying the end-vertices of the path \(P_m\) with any two vertices of \(K_{2,2,2}\), \(K_{2,2,2}-e\) and \(K_{2,2,2}-2e\), respectively, where \(e\) and \(2e\) are, respectively, an edge or any two edges of \(K_{2,2,2}\). As a by-product of this, we obtain some families of chromatically unique and chromatically equivalent classes of graphs.''