Circular choosability of graphs

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) of a graph and prove that they are equivalent. Then we prove that for any graph , . Examples are given to show that this bound is sharp in the sense that for any , there is a graph with . It is also proved that -degenerate graphs have . This bound is also sharp: for each , there is a -degenerate graph with . This shows that could be arbitrarily larger than . Finally we prove that if has maximum degree , then .