Packing in Carbon Nanotubes

Abstract

Structures realized by arrangements of regular hexagons in the plane are of interest in the chemistry of benzenoid hydrocarbons, where perfect matchings correspond to Kekulé structures which feature in the calculation of molecular energies associated with benzenoid hydrocarbon molecules. Mathematically, assembling in predictable patterns is equivalent to packing in graphs. An $H$-packing of a graph $G$ is a set of vertex-disjoint subgraphs of $G$, each of which is isomorphic to a fixed graph $H$. If $H$ is the complete graph $K_2$, the maximum $H$-packing problem becomes the familiar maximum matching problem. In this paper, we find an $H$-packing of an armchair carbon nanotube with $H$ isomorphic to $P_4$, \emph{1, 4-dimethyl cyclohexane}, and $C_6$. Further, we determine the $H$-packing of a zigzag carbon nanotube with $H$ isomorphic to \emph{1, 4-dimethyl cyclohexane}.