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H. Hilton, An Introduction to the Theory of Groups of Finite Order, Clarendon Press, ... R.P. Stanley, Combinatorics and Commutative Algebra, Birkhaüser Boston, ...

c Birkh¨auser 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 [email protected] 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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