On the genus of the complete tripartite graph $K_{n,n,1}$

TL;DR

This paper determines the genus of the complete tripartite graph $K_{n,n,1}$ for even $n$, linking it to the minimal bridges for constructing a lane-changing-free $n$-way road interchange, with novel theoretical and practical insights.

Contribution

It provides a precise formula for the genus of $K_{n,n,1}$ for even $n$, and reveals a new connection to modeling complex road intersections.

Findings

Genus of $K_{n,n,1}$ is $ ceil (n-1)(n-2)/4 ceil$ for even $n$

Establishes a link between graph genus and road interchange design

Introduces a new theoretical approach to intersection modeling

Abstract

For even $n$ we prove that the genus of the complete tripartite graph $K_{n,n,1}$ is $\lceil (n-1) (n-2)/4 \rceil$. This is the least number of bridges needed to build a complete $n$-way road interchange where changing lanes is not allowed. Both the theoretical result, and the surprising link to modelling road intersections are new.