What is it about?

We find all posible maps on surfaces whose underlying graphs are the so-called Rose Window graphs and that satisfy that the symmetry group of the map can send any incident pair vertex-edge to any other incident pair vertex-edge. In other words, the maps that we study are arc transitive and have a Rose Window graph underlying them. Arc transitive maps have at most two flag orbits. In this paper we focus on the case of exactly two orbits, as the one orbit one was studied before. To do this, we investigate the connection of these maps to consistent cycles of the underlying graph with special emphasis on such maps of smallest possible valence, namely 4. We then give a complete classification of such maps whose underlying graphs are arc-transitive Rose Window graphs.

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Why is it important?

Arc transitive maps have a large degree of symmetry; regular and chiral maps are examples of them. While regular and chiral maps have been studied in great detail, other types of arc transitive maps not so much. Here we give a families of examples to contribute to such study.

Perspectives

Highly symmetric objects are of great interest not only in mathematics. To understand them in depth we need to have as many examples as possible in mind. A further study is to classify all highly symmetric objects with certain properties. We do that here and would like to keep doing it in many other contexts.

Isabel Hubard
Instituto de Matemáticas, UNAM

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This page is a summary of: Arc‐transitive maps with underlying Rose Window graphs, Journal of Graph Theory, July 2020, Wiley, DOI: 10.1002/jgt.22608.
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