Fullerenes with the maximum Clar number

The Clar number of a fullerene is the maximum number of mutually resonant disjoint hexagons in the fullerene. It is known that the Clar number of a fullerene with nn vertices is bounded above by ⌊n/6⌋−2⌊n/6⌋−2, where ⌊x⌋⌊x⌋ represents the largest integer not greater than xx. We show that there are no fullerenes with n≡2(mod6) vertices attaining this bound. In other words, the Clar number for a fullerene with n≡2(mod6) vertices is bounded above by ⌊n/6⌋−3⌊n/6⌋−3. Moreover, we show that two experimentally produced fullerenes C80:1(D5dD5d) and C80:2(D2D2) attain the bound ⌊n/6⌋−3⌊n/6⌋−3. Finally, we present a graph-theoretical characterization for fullerenes, whose order nn is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number ⌊n/6⌋−3⌊n/6⌋−3 (respectively, ⌊n/6⌋−2⌊n/6⌋−2).