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MATHEMATICA BOHEMICA, Vol. 128, No. 4, pp. 337-347 (2003)
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The crossing number of the generalized

Petersen graph $P[3k,k]$

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Stanley Fiorini, John Baptist Gauci

* Stanley Fiorini*, * John Baptist Gauci*, Department of Mathematics, University of Malta, Msida, Malta, e-mail: ` stanley.fiorini@um.edu.mt, johnbg@waldonet.net.mt`

**Abstract:** Guy and Harary (1967) have shown that, for $k\geq3$, 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 $k\geq2.$ 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 $k\geq4$. We extend this result by showing that $P[3k,k]$ also has crossing number $k$ for all $k\geq4$.

**Keywords:** graph, drawing, crossing number, generalized Petersen graph, Cartesian product

**Classification (MSC2000):** 05C10

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