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# Article

 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 .

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