Heights of minor faces in 3-polytopes
O. V. Borodin,
A. O. Ivanova Sobolev Institute of Mathematics, Novosibirsk, Russia
Abstract:
Each 3-polytope has obviously a face
$f$ of degree
$d(f)$ at most 5 which is called
minor. The height
$h(f)$ of
$f$ is the maximum degree of the vertices incident with
$f$. A type of a face
$f$ is defined by a set of upper constraints on the degrees of vertices incident with
$f$. This follows from the double
$n$-pyramid and semiregular
$(3,3,3,n)$-polytope,
$h(f)$ can be arbitrarily large for each
$f$ if a 3-polytope is allowed to have faces of types
$(4,4,\infty)$ or
$(3,3,3,\infty)$ which are called
pyramidal. Denote the minimum height of minor faces in a given 3-polytope by
$h$. In 1996, Horňák and Jendrol' proved that every 3-polytope without pyramidal faces satisfies
$h\le39$ and constructed a 3-polytope with
$h=30$. In 2018, we proved the sharp bound
$h\le30$. In 1998, Borodin and Loparev proved that every 3-polytope with neither pyramidal faces nor
$(3,5,\infty)$-faces has a face
$f$ such that
$h(f)\le20$ if
$d(f)=3$, or
$h(f)\le11$ if
$d(f)=4$, or
$h(f)\le5$ if
$d(f)=5$, where bounds 20 and 5 are best possible. We prove that every 3-polytope with neither pyramidal faces nor
$(3,5,\infty)$-faces has
$f$ with
$h(f)\le20$ if
$d(f)=3$, or
$h(f)\le10$ if
$d(f)=4$, or
$h(f)\le5$ if
$d(f)=5$, where all bounds 20, 10, and 5 are best possible.
Keywords:
graph, plane graph, 3-polytope, structural properties, minor face, degree, height, weight.
UDC:
519.17
MSC: 35R30 Received: 31.08.2020
Revised: 14.11.2020
Accepted: 18.11.2020
DOI:
10.33048/smzh.2021.62.202