Orderings of uniquely colorable hypergraphs

For a mixed hypergraphH= (X,C,D), where C andD are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’such that everyC ∈ Cmeets some class in more than one vertex, and everyD ∈ D has a nonempty intersection with at least two classes.A vertex-order x1, x2, . . . , xn onX (n=|X|) is uniquely colorable if the subhypergraph induced by {xj : 1 j i} has precisely one coloring, for each i (1 i n). We prove that it is NP-complete to decide whether a mixed hypergraph admits a uniquely colorable vertex-order, even if the input is restricted to have just one coloring. On the other hand, via a characterization theorem it can be decided in linear time whether a given color-sequence belongs to a mixed hypergraph in which the uniquely colorable vertex-order is unique. © 2007 Elsevier B.V. All rights reserved.