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Planar-Toroidal Decomposition of $K_{12}$

Allan Bickle, Russell Campbell

TL;DR

This work resolves whether $K_{12}$ can be decomposed into a planar and a toroidal graph by combining three theoretical filters—vertex-degree constraints, triangle-count identities, and separating-set analysis—with an extensive computer search over all maximal planar graphs of order $12$. The theoretical components substantially constrain candidate pairs $(G,\overline{G})$, while the computational phase exhaustively tests torus-embeddability of complements (and near-complements) to certify nonexistence of a decomposition; the search confirms that no decomposition exists and identifies $123$ unique near-miss pairs with exactly two fewer edges in the toroidal part. The results validate the claimed nonexistence for $n=12$ and illustrate a practical methodology for similar bi-embeddability edge-partition problems, supported by publicly available data and tools. Overall, the paper settles the $N(0,1)$ case at $n=12$ and contributes a replicable framework for investigating planar and toroidal bi-embeddings in complete graphs.

Abstract

In 1978, Anderson and White asked whether there is a decomposition of $K_{12}$ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if $G$ is planar of order 12 and $H\subseteq\overline{G}$ is toroidal, then $H$ has at least two fewer edges than $\overline{G}$. A computer search found all 123 unique pairs $\left(G,H\right)$ that make this an equality.

Planar-Toroidal Decomposition of $K_{12}$

TL;DR

This work resolves whether can be decomposed into a planar and a toroidal graph by combining three theoretical filters—vertex-degree constraints, triangle-count identities, and separating-set analysis—with an extensive computer search over all maximal planar graphs of order . The theoretical components substantially constrain candidate pairs , while the computational phase exhaustively tests torus-embeddability of complements (and near-complements) to certify nonexistence of a decomposition; the search confirms that no decomposition exists and identifies unique near-miss pairs with exactly two fewer edges in the toroidal part. The results validate the claimed nonexistence for and illustrate a practical methodology for similar bi-embeddability edge-partition problems, supported by publicly available data and tools. Overall, the paper settles the case at and contributes a replicable framework for investigating planar and toroidal bi-embeddings in complete graphs.

Abstract

In 1978, Anderson and White asked whether there is a decomposition of into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if is planar of order 12 and is toroidal, then has at least two fewer edges than . A computer search found all 123 unique pairs that make this an equality.

Paper Structure

This paper contains 6 sections, 8 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theoretical Approaches
  3. Vertex Degrees
  4. Counting Triangles
  5. Separating Sets
  6. Computer Search

Key Result

Theorem 1.2

We have $\theta\left(K_{n}\right)=\left\lfloor \frac{n+7}{6}\right\rfloor$ unless $n\in\left\{ 9,10\right\}$, and $\theta\left(K_{9}\right)=\theta\left(K_{10}\right)=3$.

Figures (2)

  • Figure 1: A planar triangulation and its complement embedded on the torus with two omitted edges shown as dotted curves.
  • Figure 2: A planar triangulation and its complement embedded on the torus with two omitted edges shown as dotted curves.

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Conjecture 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 9 more