Edge-Disjoint Max Outerplanar Graphs

• The paper introduces explicit constructions that partition graph edges into outerplanar layers with tight bounds and optimal outerthickness.
• Methodologies employ combinatorial techniques and extension lemmas to increase vertex counts while preserving maximal outerplanarity.
• Results provide improved degree bounds and comparisons to 1-planar graphs, establishing a robust framework for extremal graph decomposition.

Edge-disjoint maximal outerplanar graphs constitute a sharp combinatorial framework for partitioning the edges of a graph into structurally rich outerplanar layers, with ramifications for outerthickness parameters, degree constraints, and extremal graph decompositions. Maximal outerplanar graphs are uniquely determined by their edge-count and outer cycles and serve as foundational objects in partitioning edge sets of larger graphs without redundancy or loss of maximality. Recent work provides explicit constructions and tight bounds for such edge-disjoint decompositions, culminating in the characterization of optimal outerthickness-$t$ graphs on arbitrary numbers of vertices greater than or equal to $4t$ (Okada et al., 9 Jan 2026).

1. Fundamental Definitions and Properties

A graph $G$ is outerplanar if it admits a plane embedding with all vertices residing on the boundary of the unbounded face and with no edge crossings. Outerplanar graphs are maximal if no further edge can be added without violating outerplanarity; such graphs on $n$ vertices possess exactly $2n-3$ edges and admit a unique Hamiltonian cycle on their outer face, known as the outer cycle.

The outerthickness $\theta_o(G)$ of a graph $G$ is the smallest integer $t$ such that $E(G)$ can be partitioned into $t$ edge-disjoint outerplanar graphs. Formally, for $\theta_o(G)=t$, it necessarily holds that $|E(G)|\leq t(2|V(G)|-3)$ due to the count of edges in each maximal outerplanar component.

A graph $G$ with $\theta_o(G)=t$ is maximal outerthickness-$t$ if the addition of any edge increases $\theta_o(G)$. It is optimal outerthickness-$t$ if $G$ achieves the upper bound $|E(G)|=t(2n-3)$. Optimality is equivalent to $G$ admitting a decomposition into $t$ maximal outerplanar spanning subgraphs.

2. Main Existence Theorem and Lower Bound

The central existence result establishes that for every $t\in\mathbb{N}_{>0}$ and every integer $n\geq 4t$, an optimal outerthickness-$t$ graph $G$ on $n$ vertices exists such that $\theta_o(G)=t$ and $|E(G)|=t(2n-3)$. This result is achieved by explicit combinatorial construction for $n=4t$ and an extension method for larger $n$.

The lower bound $n\ge4t$ is necessary: Considering the total edge count in $t$ maximal outerplanar graphs ($t(2n-3)$), any simple $n$-vertex graph obeys $|E(G)|\leq\binom{n}{2}$. Hence,

$t(2n-3)\leq \frac{n(n-1)}{2}$

which, by quadratic analysis, restricts solutions to $n\ge4t$.

3. Explicit Constructions for Edge-Disjoint Maximal Outerplanar Graphs

General Construction for Arbitrary $t$

For every $t\geq1$, $t$ edge-disjoint maximal outerplanar graphs are constructed on $4t$ vertices, extending methodologies by Guy and Nowakowski (1990). The construction proceeds as follows:

• Vertices are labeled $V=[4t]_0$.
• For each $i\in[0,t-1]$, vertices $i,i+t,i+2t,i+3t$ form the corners of a convex quadrilateral, joined cyclically.
• From each cornerstone, fans of edges are incident to intermediate vertices so that each quadrant maintains monotonic edge stacks.
• This forms $8t-4$ edges per layer with maximum degree $t+2$.
• Adding the diagonal $\{i, i+2t\}$ in every layer ensures maximality, advancing each graph to $8t-3=2n-3$ edges and degree $t+3$.
• All $t$ layers are edge-disjoint due to distinct fan structures in different layers.

Power-of-Two Construction with Logarithmic Degree

When $t=2^s$ for $s\geq0$, a family of $2^s$ edge-disjoint maximal outerplanar graphs can be built on $4\cdot2^s$ vertices, each with maximum degree $2s+3=O(\log n)$:

• Vertices indexed by $V=[2^{s+2}]_0\cong\mathbb{Z}_{2^{s+2}}$.
• Outer cycles are “step-$d$” with $d$ odd; i.e., $(0, d, 2d, \dots, (n-1)d, 0)$ in $\mathbb{Z}_n$.
• Base case $s=0$: $n=4$, single graph with cycle $(0,1,2,3,0)$.
• Induction: From each step-$d$ graph at $s$, two new layers on $2^{s+3}$ vertices are formed by doubling labels and by doubling-plus-one. New vertices are connected by subdividing outer edges into triangles.
• Each inductive step increases maximum degree by 2, yielding $2s+3$ for $s$ layers.

The power-of-two construction achieves improved degree bounds and may be significant in analyzing the outerthickness of complete graphs, particularly for $n\equiv3\pmod4$.

4. Extension Lemma and Vertex Growth Mechanism

The extension mechanism ensures that, for $t$ edge-disjoint maximal outerplanar graphs on $n\ge4t$ vertices, one can increase the number of vertices without compromising the edge-disjoint or maximal properties. Specifically:

• In each outerplanar layer, select a pairwise-disjoint outer-cycle edge $\{u_i,v_i\}$.
• Introduce a new vertex $x$ and, in each layer $i$, replace $\{u_i,v_i\}$ by the triangle $u_ixv_i$.
• Edge-disjointness is preserved since new triangles do not introduce overlap, and each layer remains maximal outerplanar.

By repeated application, optimal outerthickness-$t$ graphs exist for all $n\ge4t$.

5. Corollaries and Consequences

• There exist graphs that are maximal outerthickness-$t$ but not optimal. For each $t\ge2$, the complete graphs $K_{4t-4}, K_{4t-3}, K_{4t-2}$ are maximal outerthickness-$t$ yet have fewer than $t(2n-3)$ edges.
• Outerthickness-2 graphs can possess more edges than any 1-planar graph on the same number of vertices. For $n$ vertices, optimal outerthickness-2 graphs have $4n-6$ edges, whereas the maximal for 1-planar graphs is $4n-8$. Thus, there exists an infinite family of outerthickness-2 graphs that are not 1-planar.
• Degree bounds: The general construction produces degree $\Delta=n/4+3$ (linear in $n$), while the power-of-two construction achieves $\Delta=2\log_2 t+3=O(\log n)$. The latter may provide useful structural constraints in future resolutions of the outerthickness of complete graphs.

6. Context, Open Problems, and Relevance

These results generalize prior work on maximal outerplanar partitions, notably the construction techniques by Guy and Nowakowski (1990), and strengthen the theoretical framework for outerthickness decompositions. The explicit constructions with provably tight bounds underpin systematic studies of edge partitioning and inform extremal properties of dense graphs.

A salient open problem is establishing the outerthickness of all complete graphs, especially for orders congruent to $3\pmod4$, where the finer degree constraints in the power-of-two constructions may prove pivotal. The separation from 1-planarity yields new infinite families for comparison of topological and planar graph parameters.

A plausible implication is that further optimization in edge-disjoint maximal outerplanar decompositions and degree reduction may directly influence characterizations of extremal planar partitions and graph coloring thresholds. The established tight lower bound $n\ge4t$ and explicit layer growth yield a robust groundwork for related combinatorial expansion techniques and Ramsey-type results.