Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 519–536 Chamfering operation on k-orbit maps * Mar´ ıa del R´ ıo Francos Institute of Mathematics Physics and Mechanics, University of Ljubljana, Slovenia, Jadranska 19, Ljubljana 1000, Slovenia Received 20 September 2013, accepted 25 September 2014, published online 6 October 2014 Abstract A map, as a 2-cell embedding of a graph on a closed surface, is called a k-orbit map if the group of automorphisms (or symmetries) of the map partitions its set of flags into k orbits. Orbani´ c, Pellicer and Weiss studied the effects of operations as medial and trun- cation on k-orbit maps. In this paper we study the possible symmetry types of maps that result from other maps after applying the chamfering operation and we give the number of possible flag-orbits that has the chamfering map of a k-orbit map, even if we repeat this operation t times. Keywords: Map, flag graph, symmetry type graph, chamfering operation. Math. Subj. Class.: 52B15, 05C10, 57M15, 51M20, 52B10 1 Introduction Topologically, a map M is a cellular embedding of a connected graph on a closed surface, with no boundary. While combinatorially, we define a map by an edge coloured cubic graph G M , to which we refer as the flag graph of the map M, as Lins and Vince (1982-83) define it in [18] and [25], respectively. The vertex set of G M is the set of flags of the map, and the edges define the connectivity between pairs of flags. Flags are a very important tool in describing combinatorially the structure of a map. They have been used not only for maps but also for hypermaps [9, 23], maps on the surfaces with boundary [1], abstract polytopes [22] or maniplexes [28]. A map M is called a k-orbit map if its group of automorphisms, or symmetries, par- titions the set of flags into exactly k orbits. The most symmetric maps are well known as regular (or reflexible) maps, those for which its automorphism group acts transitively on their set of flags, i.e. they have exactly one flag-orbit. Other highly symmetric type of maps * This paper is a part of Bled’11 Special Issue. E-mail address: maria.delrio@fmf.uni-lj.si (Mar´ ıa del R´ ıo Francos) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/