Thickness and Outerthickness for Embedded Graphs

TL;DR

This paper improves bounds on the thickness and outerthickness of graphs based on their genus, using embedding methods, and determines the maximum outerthickness for toroidal and double torus graphs.

Contribution

It provides new upper bounds for thickness and outerthickness related to graph genus and establishes the maximum outerthickness for graphs on the torus and double torus.

Findings

Maximum outerthickness of toroidal graphs is 3.

Graphs on the double torus have thickness at most 3.

Outerthickness bounds are improved using embedding methods.

Abstract

We consider the thickness $\theta (G))$ and outerthickness $\theta _o(G)$ of a graph G in terms of its orientable and nonorientable genus. Dean and Hutchinson provided upper bounds for thickness of graphs in terms of their orientable genus. More recently, Concalves proved that the outerthickness of any planar graph is at most 2. In this paper, we apply the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness of graphs. We first obtain better upper bounds for thickness. We then use a similar approach to provide upper bounds for outerthickness of graphs in terms of their orientable and nonorientable genera. Finally we show that the outerthickness of the torus (the maximum outerthickness of all toroidal graphs) is 3. We also show that all graphs embeddable in the double torus have thickness at most 3 and outerthickness at most 5.