Bokal developed an innovative method for finding the crossing numbers of Cartesian product of two arbitrarily large graphs. In this article, the crossing number of the join product of stars and cycles are given. Afterwards, using Bokal's zip product operation, the crossing numbers of the Cartesian products of the wheel W n and all trees T with maximum degree at most five are established.

Abstract. The investigation on the crossing numbers of graphs is very difficult problem provided that an computing of the crossing number of a given graph in general is NP-complete problem. The problem of reducing the number of crossings in the graph is studied not only in the graph theory, but also by computer scientists. The exact values of the crossing numbers are known only for some graphs or some families of graphs. In the paper, we extend known results concerning crossing numbers for join products of two graphs of order five with the path P n and the cycle C n on n vertices. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path P n of length n, the crossing numbers of Cartesian products G P n for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G P n for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian product H P n is 2(n − 1). In addition, the crossing numbers of G P n for fourty graphs G on six vertices are collected.

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The exact crossing number is known only for few specific families of graphs. According to their special structure, Cartesian products of two graphs are one of few graph classes for which the exact values of crossing numbers were obtained. Let P n be a path with n + 1 vertices and P k n be the k-power of the graph P n. Very recently, some results concerning crossing numbers of P k n were obtained. For the Cartesian product of P 2 n with the cycle of length three, the value 3n − 3 for its crossing number is given. In this paper, we extend this result by proving that the crossing numbers of the Cartesian product P 2 n C 4 is 4n − 4.