All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5

Oleg V. Borodin, and Anna O. Ivanova. "All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5." Discussiones Mathematicae Graph Theory 37.1 (2017): 5-12. <http://eudml.org/doc/288017>.

@article{OlegV2017,
abstract = {Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor. We give another tight description of 3-stars in P5s: there is a vertex of degree at most 4 having three 3-neighbors. Furthermore, we show that there are only these two tight descriptions of 3-stars in P5s. Also, we give a tight description of stars with at least three rays in P5s and pose a problem of describing all such descriptions. Finally, we prove a structural theorem about P5s that might be useful in further research.},
author = {Oleg V. Borodin, Anna O. Ivanova},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {3-polytope; planar graph; structure properties; k-star; -star},
language = {eng},
number = {1},
pages = {5-12},
title = {All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5},
url = {http://eudml.org/doc/288017},
volume = {37},
year = {2017},
}

TY - JOUR
AU - Oleg V. Borodin
AU - Anna O. Ivanova
TI - All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5
JO - Discussiones Mathematicae Graph Theory
PY - 2017
VL - 37
IS - 1
SP - 5
EP - 12
AB - Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor. We give another tight description of 3-stars in P5s: there is a vertex of degree at most 4 having three 3-neighbors. Furthermore, we show that there are only these two tight descriptions of 3-stars in P5s. Also, we give a tight description of stars with at least three rays in P5s and pose a problem of describing all such descriptions. Finally, we prove a structural theorem about P5s that might be useful in further research.
LA - eng
KW - 3-polytope; planar graph; structure properties; k-star; -star
UR - http://eudml.org/doc/288017
ER -