Improper choosability of graphs embedded on the surface of genus r

A graph G is called (k,d)∗-choosable if for every assignment L satisfying |L(v)|=k for all v∈V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Let g(G) denote the girth of G and F the set of faces of G. In this paper, we prove the following results: for a graph G on surface with genus r⩾2, we have: (a) G is (2,1)∗-choosable if g(G)⩾r+9 and |F|⩾21; (b) G is (2,2)∗-choosable if g(G)⩾⌈65(r+5)⌉ and |F|⩾13; (c) G is (2,3)∗-choosable if g(G)⩾r+5 and |F|⩾14.