The Crossing Number of Cartesian Products of Stars with 5-Vertex Graphs

Abstract

Garey and Johxon have proved that the problem to determine the crossing number of graphs is NP-complete. Because of its difficulty, presently we only know the crossing number of some classes of special graphs. The crossing number of cartesian products of paths and cycles with 5-vertex graphs mostly are known, but only few cartesian products of 5-vertex graphs with star n K , 1 are known. In this paper, we extent those results,and determine the crossing numbers of cartesian products of two 5-vertex graphs with star n K , 1 .

A survey of graphs with known or bounded crossing numbers

“…Tests with QuickCross have confirmed that it is possible to find drawings with the latter number of crossings, e.g. see Figure 31 in which drawings of G 5 3 ✷S 3 and G 5 4 ✷S 3 are drawn with four crossings, rather than the five suggested by [65]. • In 2016, Hsieh and Lin [74] claimed to have determined the crossing number of the join product of various path powers with discrete graphs and path graphs.…”

“…• In 2011, He [65] considered the crossing number of G 5 3 ✷S n and G 5 4 ✷S n and claimed that both were equal to 4 n + n 2 for n ≥ 1. However, these results are contradicted by [17] and Klešč (2009) [95] respectively, who showed that each has crossing number equal to 3 n 2 n−1 2…”

On the crossing numbers of Cartesian products of stars and graphs of order six

“…For six remaining graphs G i on five vertices, the problem of estimating cr(G i S n ) is still open, even though some incorrect result were published. For example, in [8] the incorrect proof states that for the tree T on five vertices with one vertex of degree two and one vertex of degree three, the crossing number of T S n is 4⌊ n 2 ⌋⌊ n−1 2 ⌋ + ⌊ n 2 ⌋. This contradicts Corollary 9.…”

On the crossing number for Kronecker product of a tripartite graph with path

“…The Crossing number of G Â P n is known for all graphs G of order at most five, see [11,14,15,17] and for several graphs G of order six is given in [4,19]. The crossing number of G Â C n is established for all graphs G with at most four vertices in [1,10] and for some graphs on five or six vertices [6,14,18]. The crossing number of G Â S n for all graphs G of orders three or four is determined in [11,14,15] and for some graphs of order five is given in [10,16,17,20].…”

“…The crossing number of G Â C n is established for all graphs G with at most four vertices in [1,10] and for some graphs on five or six vertices [6,14,18]. The crossing number of G Â S n for all graphs G of orders three or four is determined in [11,14,15] and for some graphs of order five is given in [10,16,17,20]. Jha and Devishetty have analyzed the upper bounds for crossing number of Kronecker product of two cycles in [12].…”