The (m,k)-patch boundary code problem
Jack E. Graver · 2003
A simple (non overlapping) region of the hexagonal tessellation of the plane is uniquely determined by its boundary. This seems also to be true for “regions ” that curve around and have a simple overlap. However, Guo, Hansen and Zheng [3] constructed a pair of non isomorphic (self-overlapping) regions of the hexagonal tessellation which have the same boundary. These regions overlapped themselves several times. In this paper we prove that any region not uniquely determined by its boundary must cover some point three or more times. Benzenoid hydrocarbons or polyhexes have been studied extensively and a rich variety of mathematical questions have arisen in the course of investigating these structures. There are several different methods for coding the boundary of a polyhex, as cyclic sequences of numbers. Independent of the coding method, it is natural to ask the following two questions. Which sequences are the boundary codes of some polyhex and, and for those that are,