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

TL;DR

This paper proves that for graphs on surfaces, 3-choosability with bounded clustering is characterized by the exclusion of a specific subgraph, extending known results about planar graphs.

Contribution

It establishes a characterization of 3-choosability with bounded clustering for surface-embedded graphs based on forbidden subgraphs, generalizing previous planar graph results.

Findings

Characterization of 3-choosability with bounded clustering on surfaces.

Equivalence between excluding a specific subgraph and 3-choosability.

Extension of planar graph results to graphs on arbitrary surfaces.

Abstract

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