Optimal Outerthickness-t Graphs

• Optimal outerthickness-t graphs are graphs whose edges are partitioned into t maximal outerplanar subgraphs, exactly meeting the edge bound |E| = t(2n–3).
• They are constructed via systematic vertex insertions and cyclic labeling, ensuring each layer is maximal outerplanar with edge-disjointness and, in power-of-2 cases, low maximum degree.
• These graphs underpin foundational work in graph decomposition, aiding the separation of edge density bounds from 1-planarity and informing extremal graph theory.

An optimal outerthickness-$t$ graph is a graph $G$ on $n$ vertices whose edge set can be partitioned into $t$ edge-disjoint maximal outerplanar graphs and which attains the edge-count bound $|E(G)| = t(2n - 3)$. Such graphs achieve the minimum possible $t$ (outerthickness) for their edge count and are foundational in the study of graph decompositions, combinatorial design, and extremal graph theory. The existence, construction, and properties of these graphs are established in explicit detail for all $t \in \mathbb{N}$ and all $n \geq 4t$ (Okada et al., 9 Jan 2026).

1. Key Definitions and Metric Bounds

• An outerplanar graph admits a plane embedding with all vertices on the outer face and without edge crossings.
• A maximal outerplanar graph on $n \geq 3$ vertices contains exactly $2n-3$ edges and a unique outer (Hamiltonian) cycle.
• The outerthickness $\tau(G)$ of a graph $G$ is the minimum integer $t$ such that $E(G)$ can be partitioned into $t$ outerplanar subgraphs:

$\tau(G) = t \implies |E(G)| \leq t(2|V(G)| - 3)$

• A graph is optimal outerthickness-$t$ if $|E(G)| = t(2n-3)$ and each layer is maximal outerplanar.
• Maximal outerthickness-$t$ requires that no edge can be added without increasing $\tau(G)$.

The bound $n \geq 4t$ is both necessary and sufficient: for $t \geq 2$, it is impossible to pack $t$ edge-disjoint maximal outerplanar graphs onto fewer than $4t$ vertices, as shown by quadratic inequality derivations.

2. Edge-Disjoint Maximal Outerplanar Packings: General Construction

For every $t \geq 1$, the following construction yields $t$ edge-disjoint maximal outerplanar graphs on $4t$ vertices:

• Vertices are labeled $V = \{0, 1, \dots, 4t - 1\}$ cyclically.
• A base outerplanar graph (following Guy & Nowakowski (1990)) arranges vertices on the sides of a square, joining corners with fans of chords to triangulate the figure.
• For each shift $i = 0,1,\dots,t-1$, translate the indices of the base graph modulo $4t$ to obtain pairwise edge-disjoint copies.
• In each shifted graph, add one diagonal $\{i, i+2t\}$ to guarantee maximality.

Each layer has maximum degree $t+3$. The union of these $t$ layers yields exactly $t(2\cdot4t-3) = t(8t-3)$ edges.

3. Extension to Arbitrary Vertex Counts $n \geq 4t$

The initial construction is bootstrapped to arbitrary sizes via repeated vertex insertions:

• For graphs $\{(V, E_i)\}_{i=0}^{t-1}$ on $n \geq 4t$ vertices, select (greedily) an edge on the outer cycle for each layer such that the chosen edges form a matching.
• Add a new vertex $x$ per increment; in each graph, replace a matched edge $\{u_i, v_i\}$ by edges $\{u_i, x\}$ and $\{v_i, x\}$, maintaining maximal outerplanarity.
• Repeating this for each new vertex, a family of $t$ edge-disjoint maximal outerplanar graphs is achieved for every $n \geq 4t$.

4. Recursive Constructions with Logarithmic Maximum Degree (Power-of-2 Case)

For $t = 2^s$, a recursive construction leverages cyclic group labeling to obtain layers with low maximum degree $O(\log n)$:

• Vertices are labeled by $Z_n$ for $n = 4\cdot2^s$.
• Outer cycles use odd generators of the group.
• Inductive doubling and chord subdivision introduce new layers while preserving edge-disjointness and outerplanarity.
• The maximum degree increases by 2 with each step, yielding degree $2s+3 = 2\log_2 t + 3$ at stage $s$.

This construction enables optimal outerthickness-$t$ graphs with maximum degree logarithmic in the number of vertices, facilitating analysis on complete graphs and potential resolution of open cases for $K_n$ with $n \equiv 3 \pmod{4}$.

5. Optimality, Tightness, and Separation from 1-Planarity

• The lower bound $n \geq 4t$ is proved tight: it is impossible to fit $t$ maximal outerplanar layers for $t \geq 2$ on fewer vertices.
• For $t \geq 1$ and $n \geq 4t$, the union of constructed edge-disjoint maximal outerplanar graphs achieves $\tau(G) = t$ and saturates the edge-count bound.

A comparison with $1$-planarity highlights separation:

• 1-planar graphs on $n$ vertices have at most $4n-8$ edges, but optimal outerthickness-$2$ graphs have $4n-6$ edges for $n \geq 8$.
• Thus, an infinite family of outerthickness-$2$ graphs is known that are not 1-planar.

6. Maximal Versus Optimal Outerthickness-$t$ Graphs

• Maximal outerthickness-$t$ graphs may have fewer than $4t$ vertices; for instance, complete graphs on $4t-4$, $4t-3$, $4t-2$ vertices can have $\tau(G) = t$ without attaining the optimal edge count.
• It follows that maximal and optimal outerthickness-$t$ are distinct notions for $t \geq 2$; optimality is equivalent to precisely hitting the edge bound, while maximality concerns augmentability without exceeding outerthickness.

7. Applications and Open Problems

• These constructions decompose $K_{4t}$ minus a perfect matching into $t$ maximal outerplanar graphs.
• The power-of-2 construction, by producing logarithmic maximum degree in layered decompositions, offers a promising avenue for analyzing the outerthickness of complete graphs, especially in arithmetic cases left open by Guy & Nowakowski’s earlier methods.
• The precise characterization of optimal outerthickness-$t$ graphs enables broader explorations in graph layering, edge-partition extremal problems, and potential algorithmic applications where maximum degree constraints are relevant.

These results establish both constructions and tightness criteria for optimal outerthickness-$t$ graphs for all $t$, providing foundational tools for subsequent work in graph decompositions and outerplanar coverings (Okada et al., 9 Jan 2026).