Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Balanced subdivisions and flips on surfaces


Authors: Satoshi Murai and Yusuke Suzuki
Journal: Proc. Amer. Math. Soc. 146 (2018), 939-951
MSC (2010): Primary 05C10; Secondary 57Q15, 52B70
DOI: https://doi.org/10.1090/proc/13775
Published electronically: October 23, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the $ 2$-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.


References [Enhancements On Off] (What's this?)

  • [Al] James W. Alexander, The combinatorial theory of complexes, Ann. of Math. (2) 31 (1930), no. 2, 292-320. MR 1502943, https://doi.org/10.2307/1968099
  • [ESZ] M. N. Ellingham, Chris Stephens, and Xiaoya Zha, The nonorientable genus of complete tripartite graphs, J. Combin. Theory Ser. B 96 (2006), no. 4, 529-559. MR 2232390, https://doi.org/10.1016/j.jctb.2005.10.004
  • [IKN] I. Izmestiev, S. Klee, and I. Novik, Simplicial moves on balanced complexes, arXiv:1512.04384.
  • [Li] W. B. R. Lickorish, Simplicial moves on complexes and manifolds, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 299-320. MR 1734414, https://doi.org/10.2140/gtm.1999.2.299
  • [KNS] Ken-ichi Kawarabayashi, Atsuhiro Nakamoto, and Yusuke Suzuki, $ N$-flips in even triangulations on surfaces, J. Combin. Theory Ser. B 99 (2009), no. 1, 229-246. MR 2467828, https://doi.org/10.1016/j.jctb.2008.06.006
  • [NSS] Atsuhiro Nakamoto, Tadashi Sakuma, and Yusuke Suzuki, $ N$-flips in even triangulations on the sphere, J. Graph Theory 51 (2006), no. 3, 260-268. MR 2202056, https://doi.org/10.1002/jgt.20132
  • [Pa1] U. Pachner, Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg 57 (1987), 69-86 (German). MR 927165, https://doi.org/10.1007/BF02941601
  • [Pa2] Udo Pachner, P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), no. 2, 129-145. MR 1095161, https://doi.org/10.1016/S0195-6698(13)80080-7
  • [RY] Gerhard Ringel and J. W. T. Youngs, Das Geschlecht des vollständigen dreifärbbaren Graphen, Comment. Math. Helv. 45 (1970), 152-158 (German). MR 0262114, https://doi.org/10.1007/BF02567322
  • [Wh] Arthur T. White, The genus of the complete tripartite graph $ K_{mn,n,n}$, J. Combinatorial Theory 7 (1969), 283-285. MR 0245470

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05C10, 57Q15, 52B70

Retrieve articles in all journals with MSC (2010): 05C10, 57Q15, 52B70


Additional Information

Satoshi Murai
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan
Email: s-murai@ist.osaka-u.ac.jp

Yusuke Suzuki
Affiliation: Department of Mathematics, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan
Email: y-suzuki@math.sc.niigata-u.ac.jp

DOI: https://doi.org/10.1090/proc/13775
Received by editor(s): January 31, 2017
Received by editor(s) in revised form: April 5, 2017, and April 12, 2017
Published electronically: October 23, 2017
Additional Notes: The first author was partially supported by KAKENHI16K05102.
The second author was partially supported by KAKENHI16K05250.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society