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Chromaticity of series-parallel graphs. (English) Zbl 0856.05035
If $$P(G, \lambda)$$ denotes the chromatic polynomial of a graph $$G$$, two graphs $$G$$ and $$H$$ are said to be $$\chi$$-equivalent, written $$G \sim H$$, if $$P(G, \lambda) = P (H, \lambda)$$. Any equivalence class induced by $$\sim$$ is called a $$\chi$$-equivalence class. A $$k$$-gon tree is obtained by edge-gluing a set of cycles each of order $$k$$ successively. A graph is called a generalised polygon tree if it is a subdivision of a $$k$$-gon tree. Alternatively, a generalised polygon tree can be viewed as a 2-connected simple series-parallel graph (sp graph). In this paper some new chromatic invariants for sp graphs are introduced and it is proved that the class of polygon trees is a $$\chi$$-equivalence class. A class of sp graphs, called $$\Theta_k$$-gons, is shown to be closed under $$\chi$$-equivalence and some classes of $$\Theta_k$$-gons are identified to possess the same property; one of them is the class of $$k$$-gon trees.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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##### References:
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