Oriented edge-transitive maps

J\'an Karab\'a\v s
University of Primorska, Koper and Matej Bel University, Banská Bystrica

Roman Nedela
Matej Bel University, Banská Bystrica

Tomaž Pisanski
University of Primorska, Koper



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.

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