c Birkhäuser Verlag, Basel, 2008
Annals of Combinatorics 12 (2008) 241-269
Annals of Combinatorics
0218-0006/08/030241-29
DOI 10.1007/s00026-008-0349-y
Polytopes of Magic Labelings of Graphs and the Faces
of the Birkhoff Polytope
Maya Mohsin Ahmed
Mechanical Engineering Department, Manipal Institute of Technology, Manipal 576 104, Karnataka, India
mayaahmed@yahoo.com
Received September 9, 2004
AMS Subject Classification: 05E
Abstract. In this article, we construct and enumerate magic labelings of graphs using Hilbert
bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. This enables us to generate and enumerate perfect matchings of a graph via magic labelings of the graph. We explore
the correspondence of magic labelings of graphs with magic squares and define polytopes of
magic labelings to give a description of the faces of the Birkhoff polytope as polytopes of magic
labelings of digraphs.
Keywords: symmetric magic squares, magic labelings of graphs, Polyhedral cones, Polytopes,
Hilbert basis, Ehrhart quasi-polynomials, Invariant rings, Birkhoff polytope
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