Abstract
For a graph G, [Formula: see text], [Formula: see text] represent the clique number and the chromatic number of G, respectively. A hereditary family [Formula: see text] of graphs is called [Formula: see text]-bounded with [Formula: see text]-binding function [Formula: see text] if [Formula: see text] for all [Formula: see text]. A result of Schiermeyer shows that the class of [Formula: see text]-free graphs has a [Formula: see text]-binding function [Formula: see text] [L. Esperet, L. Lemoine, F. Maffray and G. Morel, The chromatic number of [Formula: see text]-free graphs, Discrete Math. 313 (2013) 743–754]. In this paper, we prove that the class of [Formula: see text]-free graphs has a [Formula: see text]-binding function [Formula: see text]. For the class of [Formula: see text]-free graphs, we give a [Formula: see text]-binding function [Formula: see text].
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