The author proves the following theorem: ''Let H be a peri-condensed HS and K be a cover with maximum cardinality. Then \(H\setminus K\) has a unique 1-factor''. Here is used the following terminology: A hexagonal unit cell is a plane region bounded by a regular hexagon of side length 1. A hexagonal system (HS) is a finite connected plane graph with no cut- vertices in which every region is a hexagonal unit cell. A vertex of H lying on the boundary of the exterior region of H is called an external vertex and a vertex not being external is called an internal vertex. If H has internal vertices it is said to be peri-condensed.