Seminár z algebratickej teórie grafov - Róbert Jajcay (26.10.2018)

v piatok 26 .10.2018 o 13:30 hod. v miestnosti M/XI

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Prednášajúci: Róbert Jajcay

Názov: Generalized Cayley maps and their Petrie duals

Termín: 26.10.2018, 13:30 hod., M/XI

Abstrakt:
Due to their inherent high level of symmetry, Cayley maps constitute a very important class of maps that often turn out to be orientably regular or regular. On the other hand, many important classes of orientably-regular and regular maps are indeed Cayley maps. It is therefore natural to try to generalize this important class to a larger class that still keeps the symmetry properties of the original Cayley maps. Our starting point is the observation that (original) Cayley maps are distinguished by the existence of a group of orientation preserving automorphisms acting regularly on the set of the vertices of the map. In line with this observation, we define the generalized Cayley maps to be the orientable or non-orientable maps that admit a group of map automorphisms (not necessarily orientation preserving) that act reguarly on the vertices. Since the Petrie dual operator preserves the underlying graph as well as the automorphism group of a map, the Petrie dual of a generalized Cayley map is a generalized Cayley map. Thus, Petrie duals prove to be very useful in the context of generalized Cayley maps. We present a number of examples of generalized Cayley maps, investigate the action of the Petrie dual operator on this class, and present a number of Petrie-self-dual generalized Cayley maps.

The presented results come from joint work with Jozef Siran and Yan Wang. 

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