Bounds for variables with few occurrences in conjunctive normal forms

We investigate connections between SAT (the propositional satisfiability problem) and combinatorics, around the minimum degree of variables in various forms of redundancy-free boolean conjunctive normal forms (clause-sets). Let μvd(F ) ∈ N for a clause-set F denote the minimum variable-degree, the minimum of the number of occurrences of a variable. A central result is the upper bound σ(F ) + 1 ≤ μvd(F ) ≤ nM(σ(F )) ≤ σ(F ) + 1 + log2(σ(F )) for lean clause-sets F ∈ LEAN in dependency on the surplus σ(F ) ∈ N. Lean clause-sets, defined as having no non-trivial autarkies (partial assignments satisfying some clauses and not touching the other clauses), generalise minimally unsatisfiable clause-sets, i.e., LEAN ⊃ MU . For the surplus we have σ(F ) ≤ δ(F ) = c(F )− n(F ), using the deficiency δ(F ) of clause-sets, the difference between the number c(F ) of clauses and the number n(F ) of variables. nM(k) ∈ N is the k-th “non-Mersenne” number, skipping in the sequence of natural numbers all numbers of the form 2−1. As an application of the upper bound we obtain, that clause-sets F violating μvd(F ) ≤ nM(σ(F )) must have a non-trivial autarky, so clauses can be removed satisfiability-equivalently. We obtain a polynomial time autarky reduction, but where it is open whether such an autarky itself can be found in polynomial time. We show that the upper bound is sharp, i.e., μvd(LEANδ=k) = nM(k) for all deficiencies k ∈ N, where μvd(LEANδ=k) is the maximum of μvd(F ) over F ∈ LEANδ=k. The determination of μvd(MUδ=k) =: μnM(k) seems to be a much more involved question. We show that for k ≤ 5 we have μnM(k) = nM(k), but for k = 6 we have μnM(k) = nM(k)− 1. Moreover this correction by −1 causes further corrections by −1 for infinitely many other deficiencies, resulting in the upper-bound function nM1 : N → N, an instance of a generalised non-Mersenne function found by a novel recursion scheme. Extensive introductions, overviews, conclusions, examples and open problems are provided.