Extended core and choosability of a graph

A graph G is (a, b)-choosable if for any color list of size a associated with each vertices, one can choose a subset of b colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the (a, b)-choosability of a graph and the (a, b)-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove the (5, 2)-choosability and (7, 3)-colorability of triangle-free induced subgraphs of the triangular lattice.