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CHARACTERISATION OF GRAPHS WHICH UNDERLIE REGULAR MAPS ON CLOSED SURFACES

Published online by Cambridge University Press:  01 February 1999

A. GARDINER ,
R. NEDELA ,
J. šIRÁŇ  and
M. šKOVIERA
A. GARDINER
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT
R. NEDELA
Affiliation:
Department of Mathematics, Matej Bel University, SK-975 49 Banská Bystrica, Slovakia
J. šIRÁŇ
Affiliation:
Department of Mathematics, Faculty of Civil Engineering, Slovak Technical University, SK-813 68 Bratislava, Slovakia
M. šKOVIERA
Affiliation:
Department of Computer Science, Comenius University, Mlynská Dolina, SK-842 15 Bratislava, Slovakia
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Abstract

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 Ge of every edge e is dihedral of order 4 and the stabiliser Gv of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v. 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 Hv is the cyclic subgroup of index 2 in Gv. An analogous result is proved for orientably-regular embeddings.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 59 , Issue 1 , February 1999 , pp. 100 - 108
Copyright
The London Mathematical Society 1999

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