Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
5-connected graph; contractible subgraph; minor minimally $k$-connected
5-connected graph; contractible subgraph; minor minimally $k$-connected
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.
References:
[1] Ando, K., Qin, C.: Some structural properties of minimally contraction-critically $5$-connected graphs. Discrete Math. 311 (2011), 1084-1097. DOI 10.1016/j.disc.2010.10.022 | MR 2793219 | Zbl 1222.05128
[2] Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications. American Elsevier Publishing New York (1976). MR 0411988
[3] Fijavž, G.: Graph Minors and Connectivity. Ph.D. Thesis. University of Ljubljana (2001).
[4] Kriesell, M.: Triangle density and contractibility. Comb. Probab. Comput. 14 (2005), 133-146. DOI 10.1017/S0963548304006601 | MR 2128086 | Zbl 1059.05065
[5] Kriesell, M.: How to contract an essentially $6$-connected graph to a $5$-connected graph. Discrete Math. 307 (2007), 494-510. DOI 10.1016/j.disc.2005.09.040 | MR 2287490 | Zbl 1109.05062
[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. DOI 10.1016/0012-365X(88)90216-6 | MR 0975546
[7] Qin, C., Yuan, X., Su, J.: Triangles in contraction critical $5$-connected graphs. Australas. J. Comb. 33 (2005), 139-146. MR 2170354 | Zbl 1077.05055
[8] Tutte, W. T.: A theory of $3$-connected graphs. Nederl. Akad. Wet., Proc., Ser. A 64 (1961), 441-455. MR 0140094 | Zbl 0101.40903
Search
Partner of
