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

References 1. M. Ahmed, J. De Loera, and R. Hemmecke, Polyhedral cones of magic cubes and squares, In: Discrete and Computational Geometry: The Goodman-Pollack Festschrift, B. Aronov et al., Eds., Springer-Verlag, Berlin, (2003) pp. 25–41. 2. M. Ahmed, How many squares are there, Mr. Franklin?: constructing and enumerating Franklin squares, Amer. Math. Monthly, 111 (2004) 394–410. 3. M. Ahmed, Algebraic combinatorics of magic squares, Ph.D. Thesis, University of California, Davis, 2004. 4. N. Alon and M. Tarsi, A note on graph colorings and graph polynomials, J. Combin. Theory Ser. B 70 (1) (1997) 197–201. 5. H. Anand, V.C. Dumir, and H. Gupta, A combinatorial distribution problem, Duke Math. J. 33 (1966) 757–769. 6. W.S. Andrews, Magic Squares and Cubes, 2nd Ed., Dover, New York, 1960. 7. M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969. 8. M. Beck, The arithmetic of rational polytopes, Dissertation, Temple University, 2000. 241

242

M.M. Ahmed

9. M. Beck and D. Pixton, The Ehrhart polynomial of the Birkhoff polytope, Discrete Comput. Geom. 30 (4) (2003) 623–637. 10. M. Beck, M. Cohen, J. Cuomo, and P. Gribelyuk, The number of magic squares, cubes and hypercubes, Amer. Math. Monthly, 110 (8) (2003) 707–717. 11. L.J. Billera and A. Sarangarajan, The combinatorics of permutation polytopes, In: DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 24, Amer. Math. Soc., Providence, RI, (1996) pp. 1–23. 12. R.A. Brualdi and P. Gibson, Convex polyhedra of doubly stochastic matrices I, J. Combin. Theory Ser. A 22 (2) (1977) 194–230. 13. R.A. Brualdi and P. Gibson, Convex polyhedra of doubly stochastic matrices II, J. Combin. Theory Ser. B 22 (2) (1977) 175–198. 14. R.A. Brualdi and P. Gibson, Convex polyhedra of doubly stochastic matrices III, J. Combin. Theory Ser. A 22 (3) (1977) 338–351. 15. R.A. Brualdi, Introductory Combinatorics, 3rd Ed., Prentice hall, New Jersey, 1999. 16. L. Carlitz, Enumeration of symmetric arrays, Duke Math. J. 33 (1966) 771–782. 17. C.S. Chan and D.P. Robbins, On the volume of the polytope of doubly stochastic matrices, Experiment. Math. 8 (3) (1999) 291–300. 18. J. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida, Effective lattice point counting in rational convex polytopes, J. Symbolic Comput. 38 (4) (2004) 1273–1302. 19. F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra Appl. 25 (1979) 191–196. 20. H. Gupta, Enumeration of symmetric matrices, Duke Math. J. 35 (1968) 653–659. 21. E.Q. Halleck, Magic squares subclasses as linear Diophantine systems, Ph.D. dissertation, University of California, San Diego, 2000. 22. R. Hemmecke, On the computation of Hilbert bases of cones, In: Proceedings of First International Congress of Mathematical Software, A.M. Cohen, X.S. Gao, and N. Takayama, Eds., World Scientific Publishing, River Edge, (2002) pp. 17–19. 23. H. Hilton, An Introduction to the Theory of Groups of Finite Order, Clarendon Press, Oxford, 1908. 24. D. K¨onig, Theory of Finite and Infinite Graphs, Birkh¨auser Boston, Boston, 1990. 25. L. Lo´vasz and M.D. Plummer, Matching Theory, North-Holland Publishing Co., Amsterdam, 1986. 26. I. Pak, On the number of faces of certain transportation polytopes, European J. Combin. 21 (5) (2000) 689–694. 27. A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, Chichester, 1986. 28. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997. 29. R.P. Stanley, Combinatorics and Commutative Algebra, Birkha¨user Boston, Boston, MA, 1983. 30. R.P. Stanley, Linear Homogeneous Diophantine Equations and Magic Labelings of Graphs, Duke Math. J. 40 (1973) 607–632. 31. R.P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters and Cohen-Macaulay rings, Duke Math. J. 43 (3) (1976) 511–531. 32. B.M. Stewart, Magic graphs, Canad. J. Math. 18 (1966) 1031–1059. 33. B.M. Stewart, Supermagic complete graphs, Canad. J. Math. 19 (1967) 427–438.

Magic Labelings of Graphs

243

34. B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Vienna, 1993. 35. N.M. Thi´ery, Algebraic invariants of graphs; a study based on computer exploration, ACM SIGSAM Bulletin 34 (4) (2000) 9–20. 36. D. Wallis, Magic Graphs, Birkh¨auser, Boston, 2001.

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

References 1. M. Ahmed, J. De Loera, and R. Hemmecke, Polyhedral cones of magic cubes and squares, In: Discrete and Computational Geometry: The Goodman-Pollack Festschrift, B. Aronov et al., Eds., Springer-Verlag, Berlin, (2003) pp. 25–41. 2. M. Ahmed, How many squares are there, Mr. Franklin?: constructing and enumerating Franklin squares, Amer. Math. Monthly, 111 (2004) 394–410. 3. M. Ahmed, Algebraic combinatorics of magic squares, Ph.D. Thesis, University of California, Davis, 2004. 4. N. Alon and M. Tarsi, A note on graph colorings and graph polynomials, J. Combin. Theory Ser. B 70 (1) (1997) 197–201. 5. H. Anand, V.C. Dumir, and H. Gupta, A combinatorial distribution problem, Duke Math. J. 33 (1966) 757–769. 6. W.S. Andrews, Magic Squares and Cubes, 2nd Ed., Dover, New York, 1960. 7. M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969. 8. M. Beck, The arithmetic of rational polytopes, Dissertation, Temple University, 2000. 241

242

M.M. Ahmed

9. M. Beck and D. Pixton, The Ehrhart polynomial of the Birkhoff polytope, Discrete Comput. Geom. 30 (4) (2003) 623–637. 10. M. Beck, M. Cohen, J. Cuomo, and P. Gribelyuk, The number of magic squares, cubes and hypercubes, Amer. Math. Monthly, 110 (8) (2003) 707–717. 11. L.J. Billera and A. Sarangarajan, The combinatorics of permutation polytopes, In: DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 24, Amer. Math. Soc., Providence, RI, (1996) pp. 1–23. 12. R.A. Brualdi and P. Gibson, Convex polyhedra of doubly stochastic matrices I, J. Combin. Theory Ser. A 22 (2) (1977) 194–230. 13. R.A. Brualdi and P. Gibson, Convex polyhedra of doubly stochastic matrices II, J. Combin. Theory Ser. B 22 (2) (1977) 175–198. 14. R.A. Brualdi and P. Gibson, Convex polyhedra of doubly stochastic matrices III, J. Combin. Theory Ser. A 22 (3) (1977) 338–351. 15. R.A. Brualdi, Introductory Combinatorics, 3rd Ed., Prentice hall, New Jersey, 1999. 16. L. Carlitz, Enumeration of symmetric arrays, Duke Math. J. 33 (1966) 771–782. 17. C.S. Chan and D.P. Robbins, On the volume of the polytope of doubly stochastic matrices, Experiment. Math. 8 (3) (1999) 291–300. 18. J. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida, Effective lattice point counting in rational convex polytopes, J. Symbolic Comput. 38 (4) (2004) 1273–1302. 19. F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra Appl. 25 (1979) 191–196. 20. H. Gupta, Enumeration of symmetric matrices, Duke Math. J. 35 (1968) 653–659. 21. E.Q. Halleck, Magic squares subclasses as linear Diophantine systems, Ph.D. dissertation, University of California, San Diego, 2000. 22. R. Hemmecke, On the computation of Hilbert bases of cones, In: Proceedings of First International Congress of Mathematical Software, A.M. Cohen, X.S. Gao, and N. Takayama, Eds., World Scientific Publishing, River Edge, (2002) pp. 17–19. 23. H. Hilton, An Introduction to the Theory of Groups of Finite Order, Clarendon Press, Oxford, 1908. 24. D. K¨onig, Theory of Finite and Infinite Graphs, Birkh¨auser Boston, Boston, 1990. 25. L. Lo´vasz and M.D. Plummer, Matching Theory, North-Holland Publishing Co., Amsterdam, 1986. 26. I. Pak, On the number of faces of certain transportation polytopes, European J. Combin. 21 (5) (2000) 689–694. 27. A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, Chichester, 1986. 28. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997. 29. R.P. Stanley, Combinatorics and Commutative Algebra, Birkha¨user Boston, Boston, MA, 1983. 30. R.P. Stanley, Linear Homogeneous Diophantine Equations and Magic Labelings of Graphs, Duke Math. J. 40 (1973) 607–632. 31. R.P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters and Cohen-Macaulay rings, Duke Math. J. 43 (3) (1976) 511–531. 32. B.M. Stewart, Magic graphs, Canad. J. Math. 18 (1966) 1031–1059. 33. B.M. Stewart, Supermagic complete graphs, Canad. J. Math. 19 (1967) 427–438.

Magic Labelings of Graphs

243

34. B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Vienna, 1993. 35. N.M. Thi´ery, Algebraic invariants of graphs; a study based on computer exploration, ACM SIGSAM Bulletin 34 (4) (2000) 9–20. 36. D. Wallis, Magic Graphs, Birkh¨auser, Boston, 2001.