An optimal construction for complete graph embeddings with duals of low connectivity

TL;DR

This paper presents a construction method for embedding complete graphs with duals that have low connectivity, achieving near-minimal genus and matching known lower bounds in certain cases.

Contribution

The authors introduce an optimal construction for complete graph embeddings with duals of low connectivity, demonstrating tightness of lower bounds infinitely often.

Findings

Duals of the embeddings have a cutvertex.

Genus of embeddings is close to the minimum possible.

Lower bounds are shown to be tight infinitely often.

Abstract

We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus is close to the minimum genus of the primal graph. When the number of vertices is congruent to 5 modulo 12, we further guarantee that the dual is simple and that the genera of the resulting embeddings match a lower bound of Brinkmann, Noguchi, and Van den Camp, showing that their lower bound is tight infinitely often.