The point-thicknesses of complete multipartite graphs

TL;DR

This paper determines the point-thickness of complete multipartite graphs, including complete graphs, by partitioning vertices into subsets inducing planar subgraphs, advancing understanding of graph planarity properties.

Contribution

It provides the first exact calculations of point-thickness for all complete multipartite graphs, extending known results for complete graphs.

Findings

Point-thickness of complete multipartite graphs is explicitly determined.

Point-thickness of complete graphs is derived as a special case.

Results contribute to graph planarity and graph decomposition theory.

Abstract

The point-thickness $\theta'(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ of $G$ is partitioned such that each subset induces a planar subgraph. In this paper, we determine the point-thickness of complete multipartite graphs. As a special case, we also obtain the point-thickness of complete graphs.