Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas

Ron Aharoni*, Nathan Linial

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

100 Scopus citations

Abstract

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 A related fact is that a minimal unsatisfiable CNF formula in n variables has at least n + 1 clauses (an unpublished result of M. Tarsi.)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.

Original languageAmerican English
Pages (from-to)196-204
Number of pages9
JournalJournal of Combinatorial Theory. Series A
Volume43
Issue number2
DOIs
StatePublished - Nov 1986

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