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Characterisation of Graphs which Underlie Regular Maps on Closed Surfaces
Author(s) -
Gardiner A.,
Nedela R.,
Širáň J.,
Škoviera M.
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006851
Subject(s) - dihedral group , mathematics , combinatorics , automorphism group , dihedral angle , embedding , vertex (graph theory) , automorphism , graph , normal subgroup , group (periodic table) , discrete mathematics , chemistry , computer science , artificial intelligence , hydrogen bond , organic chemistry , molecule
It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G υ of each vertex υ is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with υ. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H υ is the cyclic subgroup of index 2 in G υ . An analogous result is proved for orientably‐regular embeddings.