Description of edges incident to $3$-faces in $3$-polytopes without adjacent $3$-faces

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 $d(u)\le i$ and $d(v)\le j$. In 1940, Lebesgue proved that every $3$p̄olytope 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 $w^*\le9$ or $w\le8$, 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.