Chromatic Classes of Certain \(2\)-connected \((n, n+ 2)\)-Graphs

Let \(P(G)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, writen \(G \sim H\), if \(P(G) = P(H)\). A graph \(G\) is chromatically unique if \(G \cong H\) for any graph H such that \(H \sim G\). Let \(\mathcal{G}\) denote the class of \(2\)-connected graphs of order n and size \(n+ 2\) which contain a \(4\)-cycle or two triangles. It follows that if \(G \in \mathcal{G}\) and \(H \sim G\),then \(H \in \mathcal{G}\). In this paper, we determine all equivalence classes in \(\mathcal{G}\) under the equivalence relation \(‘\sim’\) and characterize the structures of the graphs in each class. As a by-product of these,we obtain three new families of chromatically unique graphs.