Settling the genus of the $n$-prism

TL;DR

This paper completes the proof of Ringel's conjecture on the genus of the $n$-prism, confirming its accuracy for most cases except when n=9, and covering specific residue classes modulo 12.

Contribution

It finalizes the calculation of the genus of the $n$-prism, verifying Ringel's conjecture for all but one specific case, thus advancing topological graph theory.

Findings

Ringel's conjecture holds for n ≡ 5, 9 mod 12, except n=9.

The genus formula is confirmed for almost all n, completing previous partial results.

The case n=9 remains an exception to the conjecture.

Abstract

In a 1977 paper, Ringel conjectured a formula for the genus of the $n$-prism $K_n\times K_2$ and verified its correctness for about five-sixths of all values $n$. We complete this calculation by showing that, with the exception of $n = 9$, Ringel's conjecture is true for $n \equiv 5, 9\pmod{12}$.