Planar-Toroidal Decomposition of $K_{12}$

TL;DR

This paper proves that the complete graph K_{12} cannot be decomposed into a planar and a toroidal graph, using theoretical analysis and exhaustive computer searches, and characterizes the edge differences in related decompositions.

Contribution

It demonstrates the non-existence of a planar-toroidal decomposition of K_{12} and characterizes edge counts in related graph pairs through computational methods.

Findings

No planar-toroidal decomposition of K_{12} exists.

Identified all pairs of graphs where the complement of a planar graph is toroidal with minimal edges.

Established that such toroidal complements have at least two fewer edges than the planar graph.

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.