On Edge-Disjoint Maximal Outerplanar Graphs

TL;DR

The paper addresses the problem of decomposing graphs into edge-disjoint maximal outerplanar subgraphs, proving that for every $t$ and $n \ge 4t$ there exist $t$ edge-disjoint maximal outerplanar graphs on $n$ vertices. It provides two distinct constructions: a general one extending Guy and Nowakowski to all $t$ on $4t$ vertices (with maximum degree $t+3$) and a second construction, valid when $t$ is a power of $2$, yielding $t$ such graphs with $O(\log n)$ maximum degree. These results imply the existence of optimal outerthickness-$t$ graphs for all $t$ and all $n \ge 4t$, show the $4t$ vertex bound is tight, and separate outerthickness from 1-planarity in infinite families. The work also discusses implications for the outerthickness of complete graphs and highlights open questions about unresolved cases, such as $n \equiv 3 \bmod 4$ in the complete graph setting.

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.

Figures (6)

• Figure 1: The so-called graph zero from Guy1990. For $i \in [r]_0$, the so-called graph $i$ is a copy of graph zero obtained by increasing the label of each vertex by $i \bmod 4r$. These graphs are shown to be pairwise edge-disjoint in Guy1990 and have maximum degree $r+2$.
• Figure 2: On the left the base graph $(V^0,E^0_0)$ from the proof of \ref{['lem:optimal-ot-power-2-construct']} is depicted. The graph $(V^1,E^1_0)$ in the middle, and $(V^1,E^1_1)$ on the right, are obtained from $(V^0,E^0_0)$ as described in the proof of \ref{['lem:optimal-ot-power-2-construct']}.
• Figure 3: The inductive vertex labeling in the proof of \ref{['lem:optimal-ot-power-2-construct']}. On the left, the part of the outer cycle in $(V^s,E_k^s)$ next to $v \in V^s$ is depicted. In the middle the corresponding part of the inductively defined graph $(V^{s+1},E_k^{s+1})$ is depicted. For the graph in the middle the labels of the black vertices get doubled (modulo $2^{s+3}$) and the new vertices and edges in blue are added (for $u=2v$ in the fourth step of the construction). Analogously, on the right, the construction of $(V^{s+1},E_{2^s+k}^{s+1})$ is depicted.
• Figure 4: From left to right the graphs $(V^2,E^2_0)$, $(V^2,E^2_2)$, $(V^2,E^2_1)$, and $(V^2,E^2_3)$ from the proof of \ref{['lem:optimal-ot-power-2-construct']}. The new vertices and edges are indicated in blue.
• Figure 5: The outerplanar embedding of $(V\cup \{x\},E_i \cup \{\{u_i,x\},\{v_i,x\}\})$ described in the proof of \ref{['lem:optimal-increase-vertices']}. The edge $\{u_i,v_i\}$ is on the outer cycle of the outerplanar embedding of $(V,E_i)$. The new vertex $x$ and the new edges are indicated in blue.

Theorems & Definitions (14)

• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• lemma 4
• proof
• theorem 1
• proof