On Edge-Disjoint Maximal Outerplanar Graphs

TL;DR

This paper presents two constructions for edge-disjoint maximal outerplanar graphs on n vertices, establishing tight bounds and contributing to the understanding of outerthickness for various graphs, including complete graphs.

Contribution

It introduces two novel constructions for edge-disjoint maximal outerplanar graphs, one extending previous work and the other applicable for powers of two with bounded maximum degree.

Findings

Constructions exist for all n ≥ 4t with tight bounds.

Optimal outerthickness-t graphs are achievable for all t.

One construction yields graphs with maximum degree logarithmic in n.

Abstract

We provide two constructions for $t$ edge-disjoint maximal outerplanar graphs on every number of $n \geq 4t$ vertices. The bound on the minimum number of vertices is tight. These constructions yield the existence of optimal outerthickness-$t$ graphs for every $t \in \mathbb{N}$. While one of the constructions works for all values of $t$ and extends graphs from Guy and Nowakowski (1990), the other one holds only for powers of $2$, but yields graphs with maximum degree logarithmic in the number of vertices. Thus, the latter may be helpful in tackling the open question of determining the outerthickness of all complete graphs.