TY - JOUR
T1 - Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
AU - Aharoni, Ron
AU - Linial, Nathan
PY - 1986/11
Y1 - 1986/11
N2 - Seymour (Quart. J. Math. Oxford 25 (1974), 303-312) proved that a minimal non 2-colorable hypergraph on n vertices has at least n edges. A related fact is that a minimal unsatisfiable CNF formula in n variables has at least n + 1 clauses (an unpublished result of M. Tarsi.) The link between the two results is shown; both are given infinite versions and proved using transversal theory (Seymour's original proof used linear algebra). For the proof of the first fact we give a strengthening of König's duality theorem, both in the finite and infinite cases. The structure of minimal unsatisfiable CNF formulas in n variables containing precisely n + 1 clauses is characterised, and this characterization is given a geometric interpretation.
AB - Seymour (Quart. J. Math. Oxford 25 (1974), 303-312) proved that a minimal non 2-colorable hypergraph on n vertices has at least n edges. A related fact is that a minimal unsatisfiable CNF formula in n variables has at least n + 1 clauses (an unpublished result of M. Tarsi.) The link between the two results is shown; both are given infinite versions and proved using transversal theory (Seymour's original proof used linear algebra). For the proof of the first fact we give a strengthening of König's duality theorem, both in the finite and infinite cases. The structure of minimal unsatisfiable CNF formulas in n variables containing precisely n + 1 clauses is characterised, and this characterization is given a geometric interpretation.
UR - http://www.scopus.com/inward/record.url?scp=0040072443&partnerID=8YFLogxK
U2 - 10.1016/0097-3165(86)90060-9
DO - 10.1016/0097-3165(86)90060-9
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AN - SCOPUS:0040072443
SN - 0097-3165
VL - 43
SP - 196
EP - 204
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
IS - 2
ER -