Describing faces in 3-polytopes with no vertices of degree from 5 to 7

Abstract

It is trivial that every 3-polytope has a face of degree at most 5, called minor. Back in 1940, Lebesgue gave an approximate description of minor faces in 3-polytopes, depending on 17 main parameters. In 2002, Borodin improved Lebesgue’s description on six parameters without worsening the others and suggested to find a tight description of minor faces.So far, such a tight description has been obtained only for several restricted classes of 3-polytopes: those with minimum degree 5 (Borodin, 1989), without vertices of degree 3 (Borodin and Ivanova, 2013), for plane triangulations (Borodin et al. 2014), and without vertices of degree from 4 to 7 (Borodin et al. 2017).In this paper, we consider 3-polytopes without vertices of degree from 5 to 7. A face is of type (k1,k2,…) if the set of degrees of its incident vertices is majorized by the vector (k1,k2,…).It follows from results by Horňák and Jendrol’ (1996) that every such polytope has a face of one of the types (4,4,∞), (3,8,15), (3,9,14), (3,10,13), (3,3,3,∞), (3,3,4,11), (3,4,4,4), and (3,3,3,3,4), which improves four parameters in what can be obtained from Lebesgue’s description.We prove a tight description “ (4,4,∞), (3,8,14), (3,9,13), (3,10,12), (3,3,3,∞), (3,3,4,11), (3,4,4,4), (3,3,3,3,4)” and, in particular, give constructions (unexpected for us) showing that 13 and 11 here are best possible (constructions confirming the sharpness of the other parameters were known before).