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Chromaticity of a family of $$K_ 4$$-homeomorphs. (English) Zbl 0781.05022
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 $$a+b \neq 2$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C38 Paths and cycles
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##### References:
 [1] Chao, C.Y.; Whitehead, E.G., On chromatic equivalence of graphs, () [2] Chao, C.Y.; Zhao, L.C., Chromatic polynomials of a family of graphs, Ars combin., 15, 111-129, (1983) · Zbl 0532.05027 [3] Weiming, Li, Almost every K4 homeomorph is chromatically unique, Ars combin., 23, 13-36, (1987) · Zbl 0644.05020 [4] Whitehead, E.G.; Zhao, L.C., Chromatic uniqueness and equivalence of K4 homeomorphs, J. graph theory, 8, 355-364, (1984) · Zbl 0555.05035 [5] Xu, S., A lemma in studying chromaticity, Ars combin., 32, 315-318, (1991) · Zbl 0753.05039 [6] S. Xu, Classes of chromatically equivalent graphs and polygon trees, to appear. · Zbl 0813.05030
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