Abstract
We use the term half-regular map to describe an orientable map with an orientation preserving automorphism group that is transitive on vertices and half-transitive on darts. We present a full classification of half-regular Cayley maps using the concept of skew-morphisms. We argue that half-regular Cayley maps come in two types: those that arise from two skew-morphism orbits of equal size that are both closed under inverses and those that arise from two equal-sized orbits that do not contain involutions or inverses but one contains the inverses of the other. In addition, half-regular Cayley maps of the first type are shown to be half-edge-transitive, while half-regular Cayley maps of the second type are shown to be necessarily edge-transitive. A connection between half-regular Cayley maps and regular hypermaps is also investigated.
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The authors acknowledge the support by the project APVV 0223-10 of the Slovak Agency for Research and Development and by the project Mobility—enhancing research, science and education at the Matej Bel University, ITMS code: 26110230082, under the Operational Program Education cofinanced by the European Social Fund. The first author was also supported in part by the projects VEGA 1/1005/12, VEGA 1/0577/14, SK-CN-0009-12, and NSFC 11371307.
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Jajcay, R., Nedela, R. Half-Regular Cayley Maps. Graphs and Combinatorics 31, 1003–1018 (2015). https://doi.org/10.1007/s00373-014-1428-y
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DOI: https://doi.org/10.1007/s00373-014-1428-y