A short proof of the non-biplanarity of $K_9$

TL;DR

This paper presents a concise proof demonstrating that the complete graph with nine vertices ($K_9$) cannot be embedded in a plane with only two layers, confirming its non-biplanarity.

Contribution

The paper offers a simplified and brief proof of the non-biplanarity of $K_9$, improving clarity over previous longer proofs.

Findings

$K_9$ is not biplanar.

The proof is shorter and more accessible.

Confirms classical results with a new approach.

Abstract

Battle, Harary, and Kodama (1962) and independently Tutte (1963) proved that the complete graph with nine vertices is not biplanar. Aiming towards simplicity and brevity, in this note we provide a short proof of this claim.