On the coloration of perfect graphs

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Abstract

We consider only finite graphs, without loops. Given an undirected graph G = (V, E), a k-coloring of the vertices of G is a mapping c: V → {1, 2,..., k} for which every edge xy of G has c(x)c(y). If c(v) = i we say that v has color i. Those sets c-1(T) (i = 1,..., k) that are not empty are called the color classes of the coloring c. Each color class is clearly a stable set (i.e., a subset of vertices with no edge between any two of them), hence we will frequently view a coloring as a partition into stable sets. The graph G is called k-colorable if it admits a k-coloring, and the chromatic number of G, denoted by χ(G), is the smallest integer k such that G is k-colorable. We refer to [9, 16, 29] for general results on graph theory.