Thwart numbers of some bipartite graphs

For a simple graph G, let c( G) denote the choice number of G, and for k ~ x( G) let Ck( G) be defined as c( G) is defined, except that there are only k colors available to form the lists of colors available to the vertices. The thwart number of G, denoted thw( G), is the smallest k such that Ck( G) c( G). To put it another way, thw( G) is the smallest number x( G)) of colors you need in order to assign c( G) colors to each vertex of G in a manner so fiendishly contrived that all attempts to properly color G from these lists will be thwarted. We survey what little is known about the thwart number in general, and use earlier work on choice numbers and restricted choice numbers to obtain results on thwart numbers of bipartite graphs. For instance, we show that thw(Km,n) = m 2 for n ~ mm, provided m ~ 2, and that (m ~)(m 1) thw(Km,n) ~ (m 1)2 for (m _l)m-l (m 2)m1 :s n < mm, if m 3. We also make a start toward characterizing bipartite graphs with thwart number 3.