The thickness of cartesian product $K_n \Box P_m$

TL;DR

This paper determines the thickness of the Cartesian product of complete graphs and paths, specifically $K_n ox$ $P_m$, expanding understanding of topological invariants for these graph classes.

Contribution

It provides the first explicit calculations of the thickness for the Cartesian product of complete graphs and paths, a previously unexplored class.

Findings

Thickness of $K_n ox P_m$ is explicitly calculated.

Results extend the known classes of graphs with determined thickness.

Applications to VLSI design are implied.

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.