An analogue of Franklin’s Theorem

Back in 1922, Franklin proved that every 3-polytope P5P5 with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions.The purpose of this note is to prove that every P5P5 has a vertex of degree at most 6 adjacent to a 5-vertex and another vertex of degree at most 6, which is also tight. Moreover, we prove that there is no tight description of 3-paths in P5P5s other than these two.