  We prove a variation on Heawood list-coloring for graphs on surfaces, modeled
on Thomassen's planar 5-list-coloring theorem. For epsilon>0 define the Heawood
number to be H(epsilon)=Floor((7+Sqrt[24*epsilon+1])/2). We prove that, except
for epsilon=3, every graph embedded on a surface of Euler genus epsilon>0 with
a distinguished face F can be list-colored when the vertices of F have
(H(epsilon)-2)-lists and all other vertices have H(epsilon)-lists unless the
induced subgraph on the vertices of F contains the complete graph on
H(epsilon)-1 vertices.
