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 3)-resonant tubules. The lower bound of its Clar number of k(k 3)-resonant tubules is also given. Note that the present paper does not consider the capped species.