A new result on chromaticity of K4-homoemorphs with girth 9

For a graph G, let P(G,lambda) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent if they share the same chromatic polynomial. A graph G is chromatically unique if any graph chromatically equivalent to G is isomorphic to G. A K4-homeomorph is a subdivision of the complete graph K4. In this paper, we determine a family of chromatically unique K4 -homeomorphs which have girth 9 and has exactly one path of length 1, and give sufficient and necessary condition for the graphs in this family to be chromatically unique.