Geodesic
Local structure of 9 and 10-connected graphs
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part III, Tome 391 (2011), pp. 157-197

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In his paper R. Halin (in “Recent Progress in Combinatorics”, Academic Press, 1969) discusses, what is the constant $c_k$ such that any minimally and contraction critically $k$-connected graph has at least $c_k|V(G)|$ vertices of degree $k$. Twenty years later the exact bound for $k=4$ ($c_4=1$) was found by N. Martinov and, independently, by M. Fontet. For larger $k$ exact bounds are unknown. This paper contributes to the study of local structure of minimally and contraction critically $k$-connected graphs and lower bounds for $c_k$. It was proved that $c_k\geq\frac12$ for $k=9,10$. This result extends the sequence of the lower bounds for $c_k$ which is equal to $\frac12$ to $k=6,7,8,9,10$. 
@article{ZNSL_2011_391_a7,
     author = {S. A. Obraztsova},
     title = {Local structure of 9 and 10-connected graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {157--197},
     publisher = {mathdoc},
     volume = {391},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_391_a7/}
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S. A. Obraztsova. Local structure of 9 and 10-connected graphs. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part III, Tome 391 (2011), pp. 157-197. http://geodesic.mathdoc.fr/item/ZNSL_2011_391_a7/