The thickness of cartesian product $K_n \Box P_m$

Abstract

The thickness $\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. It is a topological invariant of a graph, which was defined by W.T. Tutte in 1963 and also has important applications to VLSI design. But comparing with other topological invariants, e.g. genus and crossing number, results about thickness of graphs are few. The only types of graphs whose thicknesses have been obtained are complete graphs, complete bipartite graphs and hypercubes. In this paper, by operations on graphs, the thickness of the cartesian product $K_n \Box P_m$, $n,m \geq 2$ are obtained.