Partial list colorings

Abstract Suppose G is an s -choosable graph with n vertices, and every vertex of G is assigned a list of t colors. We conjecture that at least (t/s)n of the vertices of G can be colored from these lists. We provide lower bounds and consider related questions. For instance, we show that if G is χ -colorable (rather than being s -choosable), then more than (1−((χ−1)/χ) t )n of the vertices of G can be colored from the lists and that this is asymptotically best possible. We include a number of open questions.