Vertex-Coloring {0,1}-Edge-Weighting is W[1]-hard parameterized by feedback vertex set size, FPT by vertex cover size (with a restriction for the pre-weighted variant), and admits XP algorithms parameterized by treewidth.
21 Carsten Thomassen, Yezhou Wu, and Cun-Quan Zhang
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The Parameterized Complexity of Vertex-Coloring Edge-Weighting
Vertex-Coloring {0,1}-Edge-Weighting is W[1]-hard parameterized by feedback vertex set size, FPT by vertex cover size (with a restriction for the pre-weighted variant), and admits XP algorithms parameterized by treewidth.