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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 196792, 11 pages
http://dx.doi.org/10.1155/2014/196792
Research Article

Clar Structure and Fries Set of Fullerenes and (4,6)-Fullerenes on Surfaces

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 23 April 2014; Accepted 16 July 2014; Published 5 August 2014

Academic Editor: Qiankun Song

Copyright © 2014 Yang Gao and Heping Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Fowler and Pisanski showed that the Fries number for a fullerene on surface Σ is bounded above by , and fullerenes which attain this bound are exactly the class of leapfrog fullerenes on surface Σ. We showed that the Clar number of a fullerene on surface Σ is bounded above by , where stands for the Euler characteristic of Σ. By establishing a relation between the extremal fullerenes and the extremal (4,6)-fullerenes on the sphere, Hartung characterized the fullerenes on the sphere for which Clar numbers attain . We prove that, for a (4,6)-fullerene on surface Σ, its Clar number is bounded above by and its Fries number is bounded above by , and we characterize the (4,6)-fullerenes on surface Σ attaining these two bounds in terms of perfect Clar structure. Moreover, we characterize the fullerenes on the projective plane for which Clar numbers attain in Hartung’s method.