Light 3-faces in 3-polytopes without adjacent triangles

Abstract

Over the last decades, a lot of research has been devoted to structural and coloring problems on plane graphs that are sparse in this or that sense.

In this note we deal with the densest among sparse 3-polytopes, namely those having no adjacent 3-cycles. Borodin (1996) proved that such 3-polytopes have 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.

By $d\left(v\right)$ denote the degree of a vertex v. An edge $e=xy$ in a 3-polytope is an $(i,j)$-edge if $d\left(x\right)\le i$ and $d\left(y\right)\le j$. 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.

We prove 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.