Orderings of uniquely colorable hypergraphs

Abstract

For a mixed hypergraph H = ( X , C , D ) , where C and D are set systems over the vertex set X , a coloring is a partition of X into ‘color classes’ such that every C ∈ C meets some class in more than one vertex, and every D ∈ D has a nonempty intersection with at least two classes. A vertex-order x 1 , x 2 , … , x n on X ( n = | X | ) is uniquely colorable if the subhypergraph induced by { x j : 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.