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Precoloring extension. III: Classes of perfect graphs. (English) Zbl 0846.05034
Summary: We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graph $$G$$ are assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring of $$G$$ with at most $$k$$ colours (for some given $$k$$)? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.
For Part I see [Discrete Math. 100, No. 1-3, 267-279 (1992; Zbl 0766.05026)], and for Part II see [Acta Math. Univ. Comen., New Ser. 62, No. 1, 1-11 (1993; Zbl 0821.05026)].

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C75 Structural characterization of families of graphs
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