Pentagon–hexagon-patches with short boundaries

Pentagon–hexagon-patches are connected bridgeless plane graphs with all bounded faces pentagons or hexagons, all interior vertices of degree 3 and all boundary vertices of degree 2 or 3. In this paper we determine the minimum and maximum possible boundary lengths min (h,p) and max (h,p) of pentagon–hexagon-patches with h hexagons and p≤6 pentagons and determine which intermediate values can occur. We show that the minimal boundary length is obtained by arranging faces in a spiral fashion starting with the pentagons, while the maximum boundary length is obtained in cases where the inner dual is a tree.