Abstract
The clique-chromatic number of a graph is the least number of colors on the vertices of the graph so that no maximal clique of size at least two is monochromatic. In 2003, Gravier, Hoang, and Maffray have shown that, for any graph $$F$$ , the class of $$F$$ -free graphs has a bounded clique-chromatic number if and only if $$F$$ is a vertex-disjoint union of paths, and they give an upper bound for all such cases. In this paper, their bounds for $$F=P_2+kP_1$$ and $$F=P_3+kP_1$$ with $$k \ge 3$$ are significantly reduced to $$k+1$$ and $$k+2$$ respectively, and sharp bounds are given for some subclasses.
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