A $$2$$-fold coloring of a graph $$G$$ assigns a set of two colors to each vertex such that adjacent vertices get disjoint sets. We get a weakened version of the Borodin-Kostochka conjecture if instead of requiring a $$(\Delta(G) - 1)$$-coloring, we require a $$2$$-fold coloring from a pot of $$2\Delta(G) - 1$$ colors. A minimal counterexample to this conjecture cannot have any of the graphs here as induced subgraphs (these are $$(2d(v) - 1 : 2)$$-choosable graphs, actually online-choosable). In the pictures, i gave different colors to the components of the complement so it is easier to see the joins. Notice how severely restricted these require neighborhoods to be for minimal counterexamples. For $$3$$-fold and $$4$$-fold coloring, things get even more restricted.