On the chromatic uniqueness of the graph \(W(n,n-2)+K_ k\)
Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are said to be chromatically equivalent if \(P(G, \lambda) = P(H, \lambda)\). A graph \(G\) is chromatically unique if \(G\) is isomorphic to \(H\) for any graph \(H\) such that \(H\) and \(G\) are chromatically equivalent. Let \(W(n,n - 2)\) denote the wheel \(W_n\) with a missing spoke, and \(W(n,n - 2) + K_k\) the join of \(W(n,n - 2)\) and the complete graph \(K_k\) which is obtained from the union of \(W(n,n - 2)\) and \(K_k\). \textit{G. L. Chia} [Discrete Math. 82, No. 2, 209-212 (1990; Zbl 0712.05025)] proved that \(W(n,n-2)\) is chromatically unique for any even \(n \geq 6\). In the present paper the authors generalize this result by showing that the joint graph \(W(n,n - 2) + K_k\) is chromatically unique for any even integer \(n \geq 6\) and integer \(k \geq 0\).
scientific article; zbMATH DE number 810258