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Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices

Authors Radoslav Fulek , Jan Kynčl

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ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Computations on matrices
Keywords
  • graph genus
  • minimum rank of a partial matrix
  • Hanani-Tutte theorem
  • graph amalgamation

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Abstract

The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. 
By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Štefankovič proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus.
We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest.

Cite As Get BibTex

Radoslav Fulek and Jan Kynčl. Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.39

Author Details

Radoslav Fulek
  • IST Austria, Am Campus 1, Klosterneuburg 3400, Austria
Jan Kynčl
  • Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

Funding

  • Fulek, Radoslav: The author gratefully acknowledges support from Austrian Science Fund (FWF): M2281-N35.
  • Kynčl, Jan: Supported by project 19-04113Y of the Czech Science Foundation (GA\texorpdfstringČ{C}R), by the PRIMUS/17/SCI/3 project of Charles University and by Charles University project UNCE/SCI/004.

Acknowledgements

We are grateful to anonymous referees for comments that helped us to improve the presentation of the results.

References

  1. A. Adrian Albert. Symmetric and alternate matrices in an arbitrary field, I. Trans. Amer. Math. Soc., 43(3):386-436, 1938. URL: http://dx.doi.org/10.2307/1990068.
  2. Dan Archdeacon. The nonorientable genus is additive. J. Graph Theory, 10(3):363-383, 1986. URL: http://dx.doi.org/10.1002/jgt.3190100313.
  3. Dan Archdeacon. The orientable genus is nonadditive. J. Graph Theory, 10(3):385-401, 1986. URL: http://dx.doi.org/10.1002/jgt.3190100314.
  4. Joseph Battle, Frank Harary, Yukihiro Kodama, and J. W. T. Youngs. Additivity of the genus of a graph. Bull. Amer. Math. Soc., 68:565-568, 1962. URL: http://dx.doi.org/10.1090/S0002-9904-1962-10847-7.
  5. André Bouchet. Orientable and nonorientable genus of the complete bipartite graph. J. Combin. Theory Ser. B, 24(1):24-33, 1978. URL: http://dx.doi.org/10.1016/0095-8956(78)90073-4.
  6. G. Cairns and Y. Nikolayevsky. Bounds for generalized thrackles. Discrete Comput. Geom., 23(2):191-206, 2000. URL: http://dx.doi.org/10.1007/PL00009495.
  7. Nir Cohen, Charles R. Johnson, Leiba Rodman, and Hugo J. Woerdeman. Ranks of completions of partial matrices. In The Gohberg anniversary collection, Vol. I (Calgary, AB, 1988), volume 40 of Oper. Theory Adv. Appl., pages 165-185. Birkhäuser, Basel, 1989. Google Scholar
  8. Éric Colin de Verdière, Vojtěch Kaluža, Pavel Paták, Zuzana Patáková, and Martin Tancer. A direct proof of the strong Hanani-Tutte theorem on the projective plane. J. Graph Algorithms Appl., 21(5):939-981, 2017. URL: http://dx.doi.org/10.7155/jgaa.00445.
  9. Chandler Davis. Completing a matrix so as to minimize the rank. In Topics in operator theory and interpolation, volume 29 of Oper. Theory Adv. Appl., pages 87-95. Birkhäuser, Basel, 1988. Google Scholar
  10. D. de Caen. The ranks of tournament matrices. Amer. Math. Monthly, 98(9):829-831, 1991. URL: http://dx.doi.org/10.2307/2324270.
  11. R. W. Decker, H. H. Glover, and J. P. Huneke. The genus of the 2-amalgamations of graphs. J. Graph Theory, 5(1):95-102, 1981. URL: http://dx.doi.org/10.1002/jgt.3190050107.
  12. R. W. Decker, H. H. Glover, and J. P. Huneke. Computing the genus of the 2-amalgamations of graphs. Combinatorica, 5(4):271-282, 1985. URL: http://dx.doi.org/10.1007/BF02579241.
  13. R. Fulek and J. Kynčl. Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4. Accepted to Combinatorica, 2017. URL: http://arxiv.org/abs/1709.00508.
  14. Radoslav Fulek and Jan Kynčl. Hanani-Tutte for approximating maps of graphs. In 34th International Symposium on Computational Geometry, volume 99 of LIPIcs. Leibniz Int. Proc. Inform., pages 39:1-39:15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.39.
  15. Radoslav Fulek and Jan Kynčl. The ℤ₂-genus of Kuratowski minors. In 34th International Symposium on Computational Geometry, volume 99 of LIPIcs. Leibniz Int. Proc. Inform., pages 40:1-40:14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.40.
  16. Jonathan L. Gross and Thomas W. Tucker. Topological graph theory. Dover Publications, Inc., Mineola, NY, 2001. Reprint of the 1987 original [Wiley, New York; MR0898434 (88h:05034)] with a new preface and supplementary bibliography. Google Scholar
  17. Haim Hanani. Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae, 23:135-142, 1934. Google Scholar
  18. D. J. Kleitman. A note on the parity of the number of crossings of a graph. J. Combinatorial Theory Ser. B, 21(1):88-89, 1976. URL: http://dx.doi.org/10.1016/0095-8956(76)90032-0.
  19. Jessie MacWilliams. Orthogonal matrices over finite fields. Amer. Math. Monthly, 76:152-164, 1969. URL: http://dx.doi.org/10.2307/2317262.
  20. Gary L. Miller. An additivity theorem for the genus of a graph. J. Combin. Theory Ser. B, 43(1):25-47, 1987. URL: http://dx.doi.org/10.1016/0095-8956(87)90028-1.
  21. Bojan Mohar. An obstruction to embedding graphs in surfaces. Discrete Math., 78(1-2):135-142, 1989. URL: http://dx.doi.org/10.1016/0012-365X(89)90170-2.
  22. Bojan Mohar and Carsten Thomassen. Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 2001. Google Scholar
  23. János Pach and Géza Tóth. Which crossing number is it anyway? J. Combin. Theory Ser. B, 80(2):225-246, 2000. URL: http://dx.doi.org/10.1006/jctb.2000.1978.
  24. Michael J. Pelsmajer, Marcus Schaefer, and Despina Stasi. Strong Hanani-Tutte on the projective plane. SIAM J. Discrete Math., 23(3):1317-1323, 2009. URL: http://dx.doi.org/10.1137/08072485X.
  25. R. Bruce Richter. On the Euler genus of a 2-connected graph. J. Combin. Theory Ser. B, 43(1):60-69, 1987. URL: http://dx.doi.org/10.1016/0095-8956(87)90030-X.
  26. Gerhard Ringel. Das Geschlecht des vollständigen paaren Graphen. Abh. Math. Sem. Univ. Hamburg, 28:139-150, 1965. URL: http://dx.doi.org/10.1007/BF02993245.
  27. Gerhard Ringel. Der vollständige paare Graph auf nichtorientierbaren Flächen. J. Reine Angew. Math., 220:88-93, 1965. URL: http://dx.doi.org/10.1515/crll.1965.220.88.
  28. Marcus Schaefer. Hanani-Tutte and related results. In Geometry - intuitive, discrete, and convex, volume 24 of Bolyai Soc. Math. Stud., pages 259-299. János Bolyai Math. Soc., Budapest, 2013. URL: http://dx.doi.org/10.1007/978-3-642-41498-5_10.
  29. Marcus Schaefer. Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algorithms Appl., 17(4):367-440, 2013. URL: http://dx.doi.org/10.7155/jgaa.00298.
  30. Marcus Schaefer and Daniel Štefankovič. Block additivity of ℤ₂-embeddings. In Graph drawing, volume 8242 of Lecture Notes in Comput. Sci., pages 185-195. Springer, Cham, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03841-4_17.
  31. Mikhail Skopenkov. On approximability by embeddings of cycles in the plane. Topology Appl., 134(1):1-22, 2003. URL: http://dx.doi.org/10.1016/S0166-8641(03)00069-5.
  32. Saul Stahl. Permutation-partition pairs: a combinatorial generalization of graph embeddings. Trans. Amer. Math. Soc., 259(1):129-145, 1980. URL: http://dx.doi.org/10.2307/1998149.
  33. Saul Stahl and Lowell W. Beineke. Blocks and the nonorientable genus of graphs. J. Graph Theory, 1(1):75-78, 1977. URL: http://dx.doi.org/10.1002/jgt.3190010114.
  34. Yongge Tian. The minimum rank of a 3× 3 partial block matrix. Linear Multilinear Algebra, 50(2):125-131, 2002. URL: http://dx.doi.org/10.1080/03081080290019531.
  35. W. T. Tutte. Toward a theory of crossing numbers. J. Combinatorial Theory, 8:45-53, 1970. URL: http://dx.doi.org/10.1016/S0021-9800(70)80007-2.
  36. H. J. Woerdeman. The lower order of lower triangular operators and minimal rank extensions. Integral Equations Operator Theory, 10(6):859-879, 1987. URL: http://dx.doi.org/10.1007/BF01196124.
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