The crossing numbers of Cartesian products of paths with 5-vertex graphs

For two graphs defined from K-5 by removing one edge or two nonadjacent edges we prove that the crossing number of Cartesian product of the graph with the path of length n is 4n and 3n-1, respectively These results complete the crossing numbers list of Cartesian products of ail 5-vertex graphs with paths. We also give a description of Cartesian products of 5-vertex graphs with cycles and stars, whose crossing numbers are known. (C) 2001 Elsevier Science B.V. All rights reserved.