The crossing numbers of products of paths and stars with 4‐vertex graphs

Abstract

In this article we determine the crossing numbers of the Cartesian products of graphs on four vertices with paths and stars.The crossing number v(G) of a graph G is the minimum number of "crossings" in any "good" drawing of G in the plane. By a drawing of G in the plane Il we mean a collection of points P in Il and open arcs A in Il -P for which there are correspondences between V and P and between

Minimum crossings in join of graphs with paths and cycles

“…Bokal in [3] confirmed the general conjecture for crossing numbers of Cartesian products of paths and stars formulated in [9]. Crossing numbers of Cartesian products of stars and paths with graphs of order at most five were studied in [9,11,12]. The table in [13] shows the summary of known crossing numbers for Cartesian products of path, cycle and star with connected graphs of order five.…”

The crossing numbers of join of the special graph on six vertices with path and cycle

“…Bokal in [3] confirmed the general conjecture for crossing numbers of Cartesian products of paths and stars formulated in [9]. Crossing numbers of Cartesian products of stars and paths with graphs of order at most five were studied in [9,11,12]. The tables in [13,14,17] show the summary of known crossing numbers for Cartesian products of path, cycle and star with connected graphs of order five.…”

On the crossing numbers of Cartesian products with trees

“…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].…”