Describing $3$-faces in $3$-polytopes without adjacent triangles

Over the last several decades, much research has been devoted to structural and coloring problems on the plane graphs that are sparse in some sense. In this paper we deal with the densest instances of sparse $3$-polytopes, namely, those without adjacent $3$-cycles. Borodin proved in 1996 that such $3$-polytope has a vertex of degree at most $4$ and, moreover, an edge with the degree-sum of its end-vertices at most $9$, where both bounds are sharp. Denote the degree of a vertex $v$ by $d(v)$. An edge $e=xy$ in a $3$-polytope is an $(i,j)$-edge if $d(x)\le i$ and $d(y)\le j$. The well-known $(3,5;4,4)$-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a $3$-vertex with a $5$-vertex. In particular, this $3$-polytope has no $3$-cycles. Recently, Borodin and Ivanova proved that every $3$-polytope with neither adjacent $3$-cycles nor $(3,5)$-edges has a $3$-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp. A $3$-face $f=(x,y,z)$ is an $(i,j,k)$-face or a face of type $(i,j,k)$ if $d(x)\le i$, $d(y)\le j$, and $d(z)\le k$. The purpose of this paper is to prove that there are precisely two tight descriptions of $3$-face-types in $3$-polytopes without adjacent $3$-cycles under the above-mentioned necessary assumption of the absence of $(3,5)$-edges; namely, $\{(3,6,7) \vee (4,4,7)\}$ and $\{(4,6,7)\}$. This implies that there is a unique tight description of $3$-faces in $3$-polytopes with neither adjacent $3$-cycles nor $3$-vertices: $\{(4,4,7)\}$.