The Existence of Unsatisfiable Formulas in k-LCNF for k>=3

A CNF formula F is linear if any distinct clauses in F contain at most one common variable. A CNF formula F is exact linear if any distinct clauses in F contain exactly one common variable. All exact linear formulas are satisfiable [4], and for the class LCNF of linear formulas, the decision problem LSAT remains NP-complete. For the subclasses LCNF of LCNF≥k, in which formulas have only clauses of length at least k, the decision problem LSAT≥k remains NP-complete if there exists an unsatisfiable formula in LCNF≥k [3,5]. Therefore, the NP-completeness of SAT for LCNF≥k (k ≥ 3) is the question whether there exists an unsatisfiable formula in LCNF≥k. In [3,5], it is shown that both LCNF≥3 and LCNF≥4 contain unsatisfiable formulas by the constructions of hypergraphs and latin squares. It leaves the open question whether for each k ≥ 5 there is an unsatisfiable formula in LCNF≥k. In this paper, we present a simple and general method to construct minimal unsatisfiable formulas in k-LCNF for each k ≥ 3.