Amenable colorings Original Research Article

We investigate graph colorings that satisfy the restraint that the color assigned to a given vertex must belong to a set of allowable colors or the restraint that it must not belong to a set of disallowed colors. Following Erdős et al. [10], we say that the graph G is k-choosable if whenever sets S(x) of size k are assigned to vertices x of G, there is an ordinary graph coloring of G so that the color assigned to x belongs to S(x). Following Brown et al. [7], we say that G is (j, m)-amenable if whenever subsets R(x) of [1, 2, …, m] of size j are assigned to vertices x of G, there is an ordinary graph coloring using colors in [1, 2, …, m] so that the color assigned to x does not belong to the set R(x). We shall observe that for any graph G and positive integer k, there is a number Jk(G) which is either a nonnegative integer or ∞, so that for any positive integer j, G is (j, j + k)-amenable if and only if j ⩽ Jk(G). It is straightforward that G is k-choosable if and only if it is k-colorable and Jk(G) = ∞. We study the problem of identifying graphs with Jk(G)=J.