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Esperet and Joret proved that planar graphs with bounded maximum degree are3-colorable with bounded clustering. Liu and Wood asked whether the conclusionholds with the assumption of the bounded maximum degree replaced by assumingthat no two vertices have many common neighbors. We answer this question inpositive, in the following stronger form: Let P''_t be the complete join of twoisolated vertices with a path on t vertices. For any surface Sigma, asubgraph-closed class of graphs drawn on Sigma is 3-choosable with boundedclustering if and only if there exists t such that P''_t does not belong to theclass.

Clustered coloring of (path+2K_1)-free graphs on surfaces