Embedding a Praeger-Xu graph into a surface

TL;DR

This paper classifies all highly symmetric graph embeddings called rotary maps on surfaces, specifically those with Praeger-Xu graphs as underlying structures, linking their classification to dihedral group representations over finite fields.

Contribution

It extends the classification of rotary maps with Praeger-Xu graphs to odd primes, establishing a correspondence with multiplicity-free dihedral group representations over finite fields.

Findings

Classified all rotary maps with Praeger-Xu graphs for odd primes.

Established a correspondence with dihedral group representations.

Extended previous work from the case p=2 to all odd primes.

Abstract

Rotary maps (orientably regular maps) are highly symmetric graph embeddings on orientable surfaces. This paper classifies all rotary maps whose underlying graphs are Praeger-Xu graphs, denoted $\operatorname{C}(p,r,s)$, for any odd prime $p$ that does not divide $r$. Our main result establishes a one-to-one correspondence between the isomorphism classes of these maps and the multiplicity-free representations of the dihedral group $\operatorname{D}_{2r}$ over the finite field $\mathbb{F}_p$. This work extends a recent classification for the case where $p=2$.