Patches with short boundaries

Abstract

In this article we prove bounds for the boundary length of patches with a given set of bounded faces. We assume that with t the number of given triangles, q the number of quadrangles, and p the number of pentagons, the curvature 3t + 2q + p is at most 6 and that at an interior vertex exactly 3 faces meet. There is no restriction on the number of faces with size 6 or larger. We prove that one gets a patch with shortest boundary if one arranges the faces in a spiral order and with increasing size. Furthermore we give explicit formulas that allow to determine all boundary lengths that occur for patches with given numbers p, q and t < 2 and no bounded face larger than 6.The patches studied in this article occur as subgraphs of 3-regular graphs in mathematics as well as models for planar polycyclic hydrocarbons in chemistry where the bounds allow to decide on the (theoretical) existence of molecules for a given chemical formula.