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 Mobius 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 (MSC 2000): 05C10
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