Geodesic
Conditions for β-perfectness
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 123-148

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A β-perfect graph is a simple graph G such that χ(G') = β(G') for every induced subgraph G' of G, where χ(G') is the chromatic number of G', and β(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily).
Keywords: chromatic number, colouring number, polynomial time 
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Keijsper, Judith; Tewes, Meike. Conditions for β-perfectness. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 123-148. http://geodesic.mathdoc.fr/item/DMGT_2002_22_1_a10/