Chromatic Equivalence Classes of Some Families of Complete Tripartite Graphs

We obtain new necessary conditions on a graph which shares the same chromatic polynomial as that of the complete tripartite graph Km,n,r . Using these, we establish the chromatic equivalence classes for K1,n,n+1 (where n ? 2). This gives a partial solution to a question raised earlier by the authors. With the same technique, we further show that Kn?3,n,n+1 is chromatically unique if n ? 5. In the more general situation, we show that if 2 ? m ? n, then Km,n,n+1 is chromatically unique if n is sufficiently large