# Oriented edge-transitive maps

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J\'an Karab\'a\v s

University of Primorska, Koper and Matej Bel University, Banská Bystrica

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Roman Nedela

Matej Bel University, Banská Bystrica

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Tomaž Pisanski

University of Primorska, Koper

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**Minisymposium:**
GENERAL SESSION TALKS

**Content:**
An oriented map $M$ is edge-transitive if its group of automorphisms, $\mathrm{Aut}\,M$, acts
transitively on the edges of the underlying graph of $M$. The group of orientation-preserving
automorphisms, $\mathrm{Aut}^+\,M$, of index at most two of $\mathrm{Aut}\,M$. It follows that
the quotient map of an edge-transitive map, $\bar{M} = M\setminus\mathrm{Aut}^+\,M$, is a map on an
(quotient) orbifold with at most two edges. We show that there are exactly 8 such quotient maps
sitting on orbifolds with at most $4$ singular points, seven are spherical and one is toroidal.
The (general) classification of edge-transitive maps is well-known, due to results of Graver and
Watkins (1997), \v{S}ir\'a\v{n}, Tucker and Watkins (1999) and recently by Orbani\'c, Pellicer,
Pisanski and Tucker (2011). The general approach is computationally difficult. The complete
classification ranges to the genus 4 (in oriented case, Orbani\'c et al.) Compared to that
method we control the genus $g$ of the underlying surface by choosing a proper $g$-admissible
orbifold rather than number of edges. Thanks to the classification of `large groups of automorphisms'
(Conder, 2011) we are able to classify oriented edge-transitive maps up to genus 101.
We discuss the relationships among general classification of edge-transitive maps and
the classification of oriented edge-transitive maps. We also consider the possibilities to
extend the classification of \emph{oriented} edge-transitive maps to general situation.