Induced H-packing k-partition problem in certain carbon based nanostructures

Nanotechnology has gained recently much attention in research to develop new carbon based materials with unique properties. It generates many new materials and devices with a wide range of applications in medicine, electronics, and computer. Carbon nanotubes (CNTs) are one of the most promising resources in the field of nanotechnology. Mathematically, assembling in predictable arrays is equivalent to packing in graphs. An H-packing of a graph G is the set of vertex disjoint subgraphs of G, each of which is isomorphic to a fixed graph H. In this paper we determine perfect and almost perfect H-packing and an induced H-packing k-partition number for Armchair carbon nanotube ACNT[n, m], Zigzag carbon nanotube ZCNT[n, m], Zigzag polyhex carbon nanotube TUHC6[2m,n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TUHC_{6}[2m,n]$$\end{document}, Boron triangular carbon nanotubes BNTt[n,m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BNT_{t}[n, m]$$\end{document}, TUC4C8(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TUC_{4}C_{8}(R)$$\end{document}, TUC4C8(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TUC_{4}C_{8}(S)$$\end{document}, HAC5C6C7[n,m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HAC_{5}C_{6}C_{7} [n, m]$$\end{document} and HAC5C7[n,m]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HAC_{5}C_{7}[n, m]$$\end{document} with H isomorphic to P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{3}$$\end{document}. Further we investicate C4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{4}$$\end{document}-packing for TUC4C8(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TUC_{4}C_{8}(R)$$\end{document}.