Эта публикация цитируется в
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Дискретная математика и математическая кибернетика
An extension of Franklin's Theorem
O. V. Borodin,
A. O. Ivanova Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
Аннотация:
Back in 1922, Franklin proved that every
$3$-polytope with minimum degree
$5$ has a
$5$-vertex adjacent to two vertices of degree at most
$6$, which is tight. This result has been extended and refined in several directions.
It is well-known that each
$3$-polytope has a vertex of degree at most
$5$, called minor vertex. A
$3$-path
$uvw$ is an
$(i,j,k)$-path if
$d(u)\le i$,
$d(v)\le j$, and
$d(w)\le k$, where
$d(x)$ is the degree of a vertex
$x$. A
$3$-path is minor
$3$-path if its central vertex is minor.
The purpose of this note is to extend Franklin' Theorem to the
$3$-polytopes with minimum degree at least
$4$ by proving that there exist precisely the following ten tight descriptions of minor
$3$-paths:
$\{(6,5,6),(4,4,9),(6,4,8),(7,4,7)\}$,
$\{(6,5,6),(4,4,9),(7,4,8)\}$,
$\{(6,5,6),(6,4,9),(7,4,7)\}$,
$\{(6,5,6),(7,4,9)\}$,
$\{(6,5,8),(4,4,9),(7,4,7)\}$,
$\{(6,5,9),(7,4,7)\}$,
$\{(7,5,7),(4,4,9),(6,4,8)\}$,
$\{(7,5,7),(6,4,9)\}$,
$\{(7,5,8),(4,4,9)\}$, and
$\{(7,5,9)\}$.
Ключевые слова:
planar graph, plane map,
$3$-polytope, structure properties, tight description, path, weight.
УДК:
519.172.2
MSC: 05C75 Поступила 27 марта 2020 г., опубликована
18 сентября 2020 г.
Язык публикации: английский
DOI:
10.33048/semi.2020.17.105