Chromaticity of a family of \(K_ 4\)-homeomorphs (Q686185)

Let \(G\) be a graph and \(P(G,\lambda)\) be its chromatic polynomial. If \(P(G,\lambda)=P(H,\lambda)\) implies \(H\) is isomorphic to \(G\), then \(G\) is said to be chromatically unique. It is well known that the cycle and the cycle with one chord, are both chromatically unique. If a cycle has two chords and they do not cross, then the graph is a polygon tree and is not chromatically unique. In this paper the chromaticity of a graph consisting of a cycle with two crossing chords is discussed and necessary and sufficient conditions are given for it to be chromatically unique. Each such a graph can be viewed as a \(K_ 4\)-homeomorph \(K_ 4(w,x,y,z,1,1)\) and it is proved that this graph is not chromatically unique if and only if it is \(K_ 4(a+2,a,2,2,1,1)\) or \(K_ 4(a+1,a+3,a,2,1,1)\) or \(K_ 4(a+2,b,a,2,1,1)\) where \(a \geq 1\), \(b \geq 1\) and \(a+b \neq 2\).