Summary: Choosability and fractional chromatic numbers
This copy was printed on October 11, 1995
A graph G is (a, b)-choosable if for any assignment of a list of a colors to each
of its vertices there is a subset of b colors of each list so that subsets corresponding
to adjacent vertices are disjoint. It is shown that for every graph G, the minimum
ratio a/b where a, b range over all pairs of integers for which G is (a, b)-choosable is
equal to the fractional chromatic number of G.
Let G = (V, E) be a graph with vertex set V and edge set E, and let L(v) be a list
of allowed colors assigned to each vertex v V . We say that G is L-list-colorable if
there exists a coloring c(v) of the vertices of G such that c(v) L(v) for all v V and
c(u) = c(v) for all edges uv E. Thus, list colorings are restricted types of proper vertex
colorings. If G is L-list-colorable for every list assignment such that |L(v)| = k for all
v V , then G is called k-choosable. The choice number, ch(G), is the smallest integer k
for which G is k-choosable.
More generally, we say that G is (a, b)-choosable for some integers a and b, a 2b > 1,