Regular self-dual and self-Petrie-dual maps of arbitrary valency

TL;DR

This paper extends the existence of regular, self-dual, and self-Petrie dual maps to all odd valencies greater than or equal to five, using algebraic number theory and group coverings.

Contribution

It generalizes previous results to include all odd valencies ≥ 5, employing algebraic number theory and group coverings for construction.

Findings

Existence of such maps for all odd valencies ≥ 5.

Construction methods using algebraic number theory and group coverings.

Application to maps on groups PSL(2,p) for prime valencies.

Abstract

The main result of D. Archdeacon, M. Conder and J. \v{S}ir\'a\v{n} [Trans. Amer. Math. Soc. 366 (2014) 8, 4491-4512] implies existence of a regular, self-dual and self-Petrie dual map of any given even valency. In this paper we extend this result to any odd valency $\ge 5$. This is done by algebraic number theory and maps defined on the groups ${\rm PSL}(2,p)$ in the case of odd prime valency $\ge 5$ and valency $9$, and by coverings for the remaining odd valencies.