We introduce a generalization of the t-way test case generation problem, where parameter t is replaced by a set $$\varPhi $$Φ of Boolean conditions on attribute values. We then present two reductions of this problem to graphs; first, to graph colouring, where we link the minimal number of tests to the chromatic number of some graph; second, to hypergraph vertex covering. This latter formalization allows us to handle problems with constraints of two kinds: those that must be true for every generated test case, and those that must be true for at least one test case. Experimental results show that the proposed solution produces test suites of slightly smaller sizes than a range of existing tools, while being more general: to the best of our knowledge, our work is the first to allow existential constraints over test cases.