# Article

 Title: The contractible subgraph of $5$-connected graphs (English) Author: Qin, Chengfu Author: Guo, Xiaofeng Author: Yang, Weihua Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 63 Issue: 3 Year: 2013 Pages: 671-677 Summary lang: English . Category: math . Summary: An edge $e$ of a $k$-connected graph $G$ is said to be $k$-removable if $G-e$ is still $k$-connected. A subgraph $H$ of a $k$-connected graph is said to be $k$-contractible if its contraction results still in a $k$-connected graph. A $k$-connected graph with neither removable edge nor contractible subgraph is said to be minor minimally $k$-connected. In this paper, we show that there is a contractible subgraph in a $5$-connected graph which contains a vertex who is not contained in any triangles. Hence, every vertex of minor minimally $5$-connected graph is contained in some triangle. (English) Keyword: 5-connected graph Keyword: contractible subgraph Keyword: minor minimally $k$-connected MSC: 05C40 MSC: 05C83 idZBL: Zbl 06282104 idMR: MR3125648 DOI: 10.1007/s10587-013-0046-9 . Date available: 2013-10-07T12:02:26Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/143483 . Reference: [1] Ando, K., Qin, C.: Some structural properties of minimally contraction-critically $5$-connected graphs.Discrete Math. 311 (2011), 1084-1097. Zbl 1222.05128, MR 2793219, 10.1016/j.disc.2010.10.022 Reference: [2] Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications.American Elsevier Publishing New York (1976). MR 0411988 Reference: [3] Fijavž, G.: Graph Minors and Connectivity. Ph.D. Thesis.University of Ljubljana (2001). Reference: [4] Kriesell, M.: Triangle density and contractibility.Comb. Probab. Comput. 14 (2005), 133-146. Zbl 1059.05065, MR 2128086, 10.1017/S0963548304006601 Reference: [5] Kriesell, M.: How to contract an essentially $6$-connected graph to a $5$-connected graph.Discrete Math. 307 (2007), 494-510. Zbl 1109.05062, MR 2287490, 10.1016/j.disc.2005.09.040 Reference: [6] Mader, W.: Generalizations of critical connectivity of graphs.Proceedings of the first Japan conference on graph theory and applications. Hakone, Japan, June 1-5, 1986. Discrete Mathematics {\it 72} J. Akiyama, Y. Egawa, H. Enomoto North-Holland Amsterdam (1988), 267-283. MR 0975546, 10.1016/0012-365X(88)90216-6 Reference: [7] Qin, C., Yuan, X., Su, J.: Triangles in contraction critical $5$-connected graphs.Australas. J. Comb. 33 (2005), 139-146. Zbl 1077.05055, MR 2170354 Reference: [8] Tutte, W. T.: A theory of $3$-connected graphs.Nederl. Akad. Wet., Proc., Ser. A 64 (1961), 441-455. Zbl 0101.40903, MR 0140094 .

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