On the existence of nanojoins with given parameters

Nanojoins are parts of large carbon molecules joining several nanotubes with the same or different parameters and chemical and electrical properties. It is known that Euler's formula implies that such nanojoins must contain faces that are not hexagons if at least three tubes are joined. As the atoms in a nanojoin are carbon atoms preferring hexagonal rings, it is normally assumed that apart from hexagons only pentagons and heptagons occur. In this paper we will give necessary and sufficient conditions for the existence of nanojoins joining nanotubes with given parameters and given numbers of pentagons and heptagons.