Regular pinched maps

Archdeacon, Dan; Bonnington, C. Paul and Širáň, Jozef (2014). Regular pinched maps. Australasian Journal of Combinatorics, 58(1) pp. 16–26.

URL: http://ajc.maths.uq.edu.au/pdf/58/ajc_v58_p016.pdf

Abstract

This paper concerns pinched surfaces, also known as pseudosurfaces. A map is a graph G embedded on an oriented pinched surface. An arc of a map is an edge of G with a fixed direction. A regular map is one with a group of orientation-preserving automorphisms that acts regularly on the arcs of a map, i.e., that acts both freely and transitively.

We study regular maps on pinched surfaces. We give a relation between a regular map on a pinched surface and a natural corresponding regular map on a surface with the pinch points pulled apart. We give several constructions for regular pinched maps and present a plethora of examples. These include strongly connected maps on pinched surfaces (those that do not have a finite set of disconnecting points), as well as examples formed by gluing other regular maps along a finite set of points.

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