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Fullerenes with the maximum Clar number
The Clar number of a fullerene is the maximum number of independent resonant
hexagons in the fullerene. It is known that the Clar number of a fullerene with
n vertices is bounded above by [n/6]-2. We find that there are no fullerenes
whose order n is congruent to 2 modulo 6 attaining this bound. In other words,
the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is
bounded above by [n/6]-3. Moreover, we show that two experimentally produced
fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a
graph-theoretical characterization for fullerenes, whose order n is congruent
to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3
(respectively, [n/6]-2)
Linear Complexity Hexahedral Mesh Generation
We show that any polyhedron forming a topological ball with an even number of
quadrilateral sides can be partitioned into O(n) topological cubes, meeting
face to face. The result generalizes to non-simply-connected polyhedra
satisfying an additional bipartiteness condition. The same techniques can also
be used to reduce the geometric version of the hexahedral mesh generation
problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at
the 12th ACM Symp. on Computational Geometry. This is the final version, and
will appear in a special issue of Computational Geometry: Theory and
Applications for papers from SCG '9
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