About vertices of degree~$k$ of minimally and contraction critically $k$-connected graphs: upper bounds
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part III, Tome 391 (2011), pp. 198-210
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In the article [4], R. Halin asked, what the constant $c_k$ such that any minimally and contraction critically $k$-connected graph has at least $c_k|V(G)|$ vertices of degree $k$. Exact bound for $k=4$ ($c_4=1$) and no upper bound for larger $k$ is known now. We found upper bounds for $c_k$ for $k\geq5$.
@article{ZNSL_2011_391_a8,
author = {S. A. Obraztsova and A. V. Pastor},
title = {About vertices of degree~$k$ of minimally and contraction critically $k$-connected graphs: upper bounds},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {198--210},
publisher = {mathdoc},
volume = {391},
year = {2011},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_391_a8/}
}
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S. A. Obraztsova; A. V. Pastor. About vertices of degree~$k$ of minimally and contraction critically $k$-connected graphs: upper bounds. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part III, Tome 391 (2011), pp. 198-210. http://geodesic.mathdoc.fr/item/ZNSL_2011_391_a8/