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Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are said to be chromatically equivalent, denoted \(G \sim H\), if \(P(G, \lambda) = P(H, \lambda)\). We write \([G] = \{H| H \sim G\}\). If \([G] = \{G\}\), then \(G\) is said to be chromatically unique. In this paper, we discuss a chromatically equivalent pair of graphs in one family of \(K_4\)-homeomorphs, \(K_4(1, 2, 8, d, e, f)\). The obtained result can be extended in the study of chromatic equivalence classes of \(K_4(1, 2, 8, d, e, f)\) and chromatic uniqueness of \(K_4\)-homeomorphs with girth \(11\).