The weight $ w(e) $ of an edge $ e $ in a $ 3 $ -polytope is the sum of the degrees of its end vertices.An edge $ e=uv $ is an $ (i,j) $ -edge if $ d(u)\leq i $ and $ d(v)\leq j $ .In 1940, Lebesgue proved that every $ 3 $ -polytope contains a $ (3,11) $ -, $ (4,7) $ -, or $ (5,6) $ -edge, where $ 7 $ and $ 6 $ are the best possible.In 1955, Kotzig proved that every $ 3 $ -polytope contains an edge whose end-vertex degrees sum to at most $ 13 $ , and this bound is sharp.Borodin (1987), answering a question by Erdős, proved that every planar graph without vertices of degree less than $ 3 $ contains such an edge.Moreover, Borodin (1991) strengthened this result by proving that there exists either a $ (3,10) $ -, $ (4,7) $ -, or $ (5,6) $ -edge.For $ 3 $ -polytopes, upper bounds were obtained for the minimal weight (the sum of the degrees of the end vertices) of all its edges, denoted by $ w $ ; of edges incident to a $ 3 $ -face, denoted by $ w^{*} $ ; and of edges incident to two $ 3 $ -faces, denoted by $ w^{**} $ .In particular, Borodin (1996) proved that if $ w^{**}=\infty $ , i.e., there are no edges incident to two $ 3 $ -faces, then either $ w^{*}\leq 9 $ or $ w\leq 8 $ , and both bounds are the best possible.Recently, we have strengthened this fact by proving that $ w^{**}=\infty $ implies the existence of either a $ (3,6) $ -edge or a $ (4,4) $ -edge incident to a $ 3 $ -face, or else a $ (3,5) $ -edge, with an exact description.(It is well known that if $ (3,5) $ -edges are present, then $ 3 $ -faces may be absent altogether.)The aim of our paper is to strengthen the above result by proving that $ w^{**}=\infty $ implies either a $ (3,6) $ -edge surrounded by a $ 3 $ -face and a $ 4 $ -face, or a $ (4,4) $ -edge surrounded by a $ 3 $ -face and a $ 7^{-} $ -face, or a $ (3,5) $ -edge, where none of the parameters can be improved.The main difficulty was to construct a $ 3 $ -polytope confirming the sharpness of $ 7 $ in this description.
