The crossing numbers of products of 5-vertex graphs with cycles

Abstract

There are known several exact results on the crossing numbers of Cartesian products of cycles with "small" graphs. In this paper we summarise known results and we give the crossing number of the Cartesian product H × C n for the specific 5-vertex graph H.

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

“…In [2,9], the crossing numbers of G C n for all graphs G of order at most four are given. In addition, the crossing numbers of G C n are known for some graphs G on five or six vertices [4,16]. Bokal in [3] confirmed the general conjecture for crossing numbers of Cartesian products of paths and stars formulated in [9].…”

Minimum crossings in join of graphs with paths and cycles

“…In [1,9], the crossing numbers of G C n for all graphs G of order at most four are established. In addition, the crossing numbers of G C n are known for some graphs G on five or six vertices [4,16]. Bokal in [3] confirmed the general conjecture for crossing numbers of Cartesian products of paths and stars formulated in [9].…”

The crossing numbers of products with cycles

“…Beineke and Ringeisen in [2] as well as Jendrol' andŠčerbová in [10] determined the crossing numbers of the Cartesian products of all graphs on four vertices with cycles. Klešč in [11], [12], [13], [14], Klešč, Richter and Stobert in [15], and Klešč and Kocúrová in [16] gave the crossing numbers of G 2C n for several graphs G of order five.…”