| Title:
|
The crossing number of the generalized Petersen graph $P[3k,k]$ (English) |
| Author:
|
Fiorini, Stanley |
| Author:
|
Gauci, John Baptist |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
128 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
337-347 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Guy and Harary (1967) have shown that, for $k\ge 3$, the graph $P[2k,k]$ is homeomorphic to the Möbius ladder ${M_{2k}}$, so that its crossing number is one; it is well known that $P[2k,2]$ is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of $P[2k+1,2]$ is three, for $k\ge 2.$ Fiorini (1986) and Richter and Salazar (2002) have shown that $P[9,3]$ has crossing number two and that $P[3k,3]$ has crossing number $k$, provided $k\ge 4$. We extend this result by showing that $P[3k,k]$ also has crossing number $k$ for all $k\ge 4$. (English) |
| Keyword:
|
graph |
| Keyword:
|
drawing |
| Keyword:
|
crossing number |
| Keyword:
|
generalized Petersen graph |
| Keyword:
|
Cartesian product |
| MSC:
|
05C10 |
| idZBL:
|
Zbl 1050.05034 |
| idMR:
|
MR2032472 |
| DOI:
|
10.21136/MB.2003.134001 |
| . |
| Date available:
|
2009-09-24T22:10:31Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134001 |
| . |
| Reference:
|
[1] Exoo G., Harary F., Kabell J.: The crossing numbers of some generalized Petersen graphs.Math. Scand. 48 (1981), 184–188. MR 0631334, 10.7146/math.scand.a-11910 |
| Reference:
|
[2] Fiorini S.: On the crossing number of generalized Petersen graphs.Ann. Discrete Math. 30 (1986), 225–242. Zbl 0595.05030, MR 0861299 |
| Reference:
|
[3] Guy R. K., Harary F.: On the Möbius ladders.Canad. Math. Bull. 10 (1967), 493–496. MR 0224499, 10.4153/CMB-1967-046-4 |
| Reference:
|
[4] Jendrol’ S., Ščerbová M.: On the crossing numbers of ${S_m}\times {C_n}$.Čas. Pěst. Mat. 107 (1982), 225–230. MR 0673046 |
| Reference:
|
[5] Kuratowski K.: Sur le problème des courbes gauches en topologie.Fund. Math. 15 (1930), 271–283. 10.4064/fm-15-1-271-283 |
| Reference:
|
[6] Richter R. B., Salazar G.: The crossing number of $P(n,3)$.Graphs Combin. 18 (2002), 381–394. MR 1913677, 10.1007/s003730200028 |
| . |
