Graph coloring satisfying restraints

For an integer k 2 2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v) # r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic number of G. For any k Z= 3, there are infinitely many amenable k-critical graphs. For k 2 3, we use a construction of B. Toft and amenable graphs to associate a k-colorable graph to any k-colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property--being strongly critical-that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Haj6s construction applied to a single edge in each of two strongly k-critical graphs yields an amenable graph. However, for any k 2 5, there are amenable k-critical graphs for which the Haj6s construction on two copies is not amenable.