k-resonance of open-ended carbon nanotubes

An open-ended carbon nanotube or a tubule is a part of some regular hexagonal tessellation of a cylinder. A tubule T is said to be k-resonant if for every k ( or fewer) pairwise disjoint hexagons, the subgraph obtained from T by deleting all the vertices of these hexagons must have a Kekule structure ( perfect matching) or must be empty. The 1-resonant tubules can be constructed by an approach provided in H. Zhang and F. Zhang, Discrete Appl. Math. 36 ( 1992) 291. In this paper, we give the construction method of k(k greater than or equal to 3)-resonant tubules. The lower bound of its Clar number of k(k greater than or equal to 3)-resonant tubules is also given. Note that the present paper does not consider the capped species.