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graph; drawing; crossing number; generalized Petersen graph; Cartesian product
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 $k\ge 4$.
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[2] Fiorini S.: On the crossing number of generalized Petersen graphs. Ann. Discrete Math. 30 (1986), 225–242. MR 0861299 | Zbl 0595.05030
[3] Guy R. K., Harary F.: On the Möbius ladders. Canad. Math. Bull. 10 (1967), 493–496. DOI 10.4153/CMB-1967-046-4 | MR 0224499
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[6] Richter R. B., Salazar G.: The crossing number of $P(n,3)$. Graphs Combin. 18 (2002), 381–394. DOI 10.1007/s003730200028 | MR 1913677
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