Previous |  Up |  Next


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:
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


Files Size Format View
CzechMathJ_63-2013-3_7.pdf 236.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo