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

On the crossing numbers of Cartesian products with trees

“…Klešč proved this conjecture for n = 4 in [14], where he also determined cr(S 4 C m ) for m ≥ 3. For general n, the conjecture was recently confirmed by the author in [7].…”

“…For general n, the conjecture was recently confirmed by the author in [7]. In [18], Klešč determined the crossing number of G P m and G S n for any graph G of order four and, in [15], the crossing number of G P m for any graph G of order five. For several graphs of order five, the crossing numbers of their Cartesian product with C n or S n are also known, as well as some other Cartesian products, most of which are due to Klešč [14][15][16][17]19].…”

“…In [18], Klešč determined the crossing number of G P m and G S n for any graph G of order four and, in [15], the crossing number of G P m for any graph G of order five. For several graphs of order five, the crossing numbers of their Cartesian product with C n or S n are also known, as well as some other Cartesian products, most of which are due to Klešč [14][15][16][17]19]. The methods rely on the high level of symmetry that these Cartesian products exhibit and mostly apply the following approach: if no copy of G has a sufficient number of crossings, then enough crossings can be found in the drawing, otherwise, the claim follows by induction.…”

The crossing number of G ▭ C n for the graph G on six vertices

“…Beineke and Ringeisen in [4], Jendrol' andŠčerbová in [9] determined the crossing numbers of the Cartesian products of all graphs on four vertices with cycles. Klešč in [10], [12], [13], [14], Klešč, Richter and Stobert in [15], and Klešč and Kocúrová in [16] gave the crossing numbers of G C n for several graphs of order five. Harary et al [8] conjectured that the crossing number of C m C n is (m − 2)n, for all m, n satisfying 3 ≤ m ≤ n. This has been proved only for m, n satisfying n ≥ m, m ≤ 7 ( [1], [2], [3], [4], [5], [15], [17], [18]).…”

“…In [12] and [14], all known values of crossing numbers for the Cartesian products of cycles and graphs of order five are presented. We are interested in the crossing numbers of Cartesian products of graphs on six vertices with cycles.…”