Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number. Key words: Hexagonal system, perfect matching, resonant set, alternating set, Clar number Preprint submitted to Elsevier 26 February 2009 

• Type: Journal
• Year: 2010
• Where: CMA
• Authors: Sandi Klavzar, Khaled Salem, Andrej Taranenko
• Keywords: Clar Number, CMA 2010, Hexagonal System, Maximal Alternating Sets