Crossing number of Cartesian product of prism and path

Abstract

An m-prism is the Cartesian product of an m-cycle and a path with 2 vertices. We prove that the crossing number of the join of an m-prism (m ! 4) and a graph with k isolated vertices is km for each k 2 f1, 2g: We then use this result to prove that the crossing number of the Cartesian product of a 5-prism and a path with n vertices is 10ðn À 1Þ: This answers partially the conjecture raised by Peng and Yiew (in 2006) in the affirmative.

Crossing Numbers of the Cartesian Product of the Double Triangular Snake Graphs With Path Pm.

“…There are few results with Cr(G × Pm), where G is a graph on 7 vertices. Finding the value of Cr(G × Pm) has been investigated in [1], [2], [3], [5], [6], [7], [8], [9], [10]. In this paper, we evaluated the exact value of Cr(DT 2 × P m ) for m ≥ 2.…”