| 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:
|
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| 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 |
| . |
