Classification of hypothetical doubly and triply periodic porous graphitic structures by tilings of neck-like units

In this study, we extend the concept of constructing high-genus fullerenes with neck-like structures to the classification of a wide family of doubly periodic graphitic structures. The neck structures are obtained by peeling the outer (positively curved) part of a toroidal carbon nanotube off, meanwhile leaving the central hole unchanged. A doubly periodic structure is then characterized by the original tiling and the shape of the necks used. Through the consideration of the coloring problem of uniform tilings, a systematic method of constructing similar structures with periodicities along all three dimensions is developed. The P type Schwarzite, extensively studied in the literature, can be classified within the scheme if the shape of the necks is carefully chosen.