| Title:
|
The basis number of some special non-planar graphs (English) |
| Author:
|
Alsardary, Salar Y. |
| Author:
|
Ali, Ali A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
225-240 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The basis number of a graph $G$ was defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. He proved that for $m,n\ge 5$, the basis number $b(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ is equal to 4 except for $K_{6,10}$, $K_{5,n}$ and $K_{6,n}$ with $n=5,6,7,8$. We determine the basis number of some particular non-planar graphs such as $K_{5,n}$ and $K_{6,n}$, $n=5,6,7,8$, and $r$-cages for $r=5,6,7,8$, and the Robertson graph. (English) |
| Keyword:
|
graphs |
| Keyword:
|
basis number |
| Keyword:
|
cycle space |
| Keyword:
|
basis |
| MSC:
|
05C35 |
| MSC:
|
05C38 |
| idZBL:
|
Zbl 1021.05053 |
| idMR:
|
MR1983447 |
| . |
| Date available:
|
2009-09-24T11:00:54Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127795 |
| . |
| Reference:
|
[1] A. A. Ali and S. Y. Alsardary: On the basis number of a graph.Dirasat (Science) 14 (1987), 43–51. |
| Reference:
|
[2] J. A. Banks and E. F. Schmeichel: The basis number of the $n$-cube.J. Combin Theory, Ser. B 33 (1982), 95–100. MR 0685059, 10.1016/0095-8956(82)90061-2 |
| Reference:
|
[3] J. A. Bondy and S. R. Murty: Graph Theory with Applications.Amer. Elsevier, New York, 1976. MR 0411988 |
| Reference:
|
[4] F. Harary: Graph Theory.2nd ed., Addison-Wesely, Reading, Massachusetts, 1971. |
| Reference:
|
[5] S. MacLane: A combinatorial condition for planar graphs.Fund. Math. 28 (1937), 22–32. Zbl 0015.37501 |
| Reference:
|
[6] N. Robertson: The smallest graph of girth 5 and valency 4.Bull. Amer. Math. Soc. 30 (1981), 824–825. MR 0167974 |
| Reference:
|
[7] E. F. Schmeichel: The basis number of a graph.J. Combin. Theory, Ser. B 30 (1981), 123–129. Zbl 0385.05031, MR 0615307, 10.1016/0095-8956(81)90057-5 |
| Reference:
|
[8] W. T. Tutte: Connectivity in Graphs.Univ. Toronto press, Toronto, 1966. Zbl 0146.45603, MR 0210617 |
| . |
