List colorings and reducibility Original Research Article

Let G be a simple graph, L(v) a list of allowed colors assigned to each vertex v of G, and U an arbitrary subset of the vertex set. The graph G is called k-choosable if, for any list assignment with ¦L(v)¦ = k for all v ϵ V, it is possible to color all vertices with colors from their lists in a proper way (i.e., no monochromatic edge occurs). We say that G is U-reducible if, for every list assignment of G where all lists have the same number of elements, we can color the vertices of U with colors from their lists such that for every vertex v ∉ U at most one color of L(v) appears in the coloring of those neighbors of v which belong to U.