Tight upper bounds for the p-anionic Clar number of fullerenes

A fullerene is an all-carbon molecule with a polyhedral structure whereeach atom is bonded to three other atoms and each face is either apentagon or a hexagon. Fullerenes correspond to 3-regular planar graphswhose faces have sizes 5 or 6. The p-anionic Clar number C_(p)(G) of afullerene G is equal to p + h, where h is maximized over all choices ofp + h independent faces (exactly p pentagons and h hexagons) thedeletion of whose vertices leave a graph with a perfect matching. Thisdefinition is motivated by the chemical observation that pentagonalrings can accommodate an extra electron, so that the pentagons of afullerene with charge −p, compete with the hexagons to host ‘Clarsextets’ of six electrons, and pentagons will preferentially acquire thep excess electrons of the anion. Tight upper bounds are established for the p-anionic Clar number offullerenes for p > 0. The upper bounds are derived via graph theoreticarguments and new results on minimal cyclic-k-edge cutsets in IPRfullerenes (fullerenes that have all pentagons pairwise disjoint). Thesebounds are shown to be tight by infinite families of fullerenes thatachieve them.