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.
@article{bwmeta1.element.doi-10_7151_dmgt_1905,
author = {Oleg V. Borodin and Anna O. Ivanova},
title = {All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {5-12},
zbl = {1354.05044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1905}
}
Oleg V. Borodin; Anna O. Ivanova. All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 5-12. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1905/