On chromatic uniqueness of \(K_3\) homeomorphs (Q2719923)

Let \(P(G,\lambda)\) be the chromatic polynomial of \(G\). For any two graphs \(G\) and \(H\), if \(P(G,\lambda)= P(H,\lambda)\), then \(G\) and \(H\) are called chromatically equivalent. If \(P(G,\lambda)= P(H,\lambda)\) also implies that \(G\) and \(H\) are isomorphic, then \(G\) is called chromatically unique. A graph is called \(K_4\) homeomorphic if it can be obtained from \(K_4\) by replacing the six edges in \(K_4\) by six paths with length \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\), respectively. The authors present necessary and sufficient conditions for two families of \(K_4\) homeomorphic graphs to be chromatically unique and chromatically equivalent.