On Realizing a Single Degree Sequence by a Bipartite Graph (Invited Paper)

Authors Amotz Bar-Noy, Toni Böhnlein, David Peleg , Dror Rawitz

Thumbnail PDF


  • Filesize: 0.91 MB
  • 17 pages

Document Identifiers

Author Details

Amotz Bar-Noy
  • City University of New York (CUNY), NY, USA
Toni Böhnlein
  • Weizmann Institute of Science, Rehovot, Israel
David Peleg
  • Weizmann Institute of Science, Rehovot, Israel
Dror Rawitz
  • Bar Ilan University, Ramat-Gan, Israel

Cite AsGet BibTex

Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On Realizing a Single Degree Sequence by a Bipartite Graph (Invited Paper). In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


This paper addresses the classical problem of characterizing degree sequences that can be realized by a bipartite graph. For the simpler variant of the problem, where a partition of the sequence into the two sides of the bipartite graph is given as part of the input, a complete characterization was given by Gale and Ryser over 60 years ago. However, the general question, in which both the partition and the realizing graph need to be determined, is still open. This paper provides an overview of some of the known results on this problem in interesting special cases, including realizations by bipartite graphs and bipartite multigraphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Degree Sequences
  • Graph Realization
  • Bipartite Graphs
  • Graphic Sequences
  • Bigraphic Sequences
  • Multigraph Realization


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Patrick Adams and Yuri Nikolayevsky. Planar bipartite biregular degree sequences. Discr. Math., 342:433-440, 2019. Google Scholar
  2. Martin Aigner and Eberhard Triesch. Realizability and uniqueness in graphs. Discr. Math., 136:3-20, 1994. Google Scholar
  3. Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On realizing even degree sequences by bipartite graphs. Unpublished manuscript, 2022. Google Scholar
  4. Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On the role of high-low partitions in realizing a degree sequence by a bipartite graph. Unpublished manuscript, 2022. Google Scholar
  5. Richard Bellman. Notes on the theory of dynamic programming iv-maximization over discrete sets. Naval Research Logistics Quarterly, 3(1-2):67-70, 1956. Google Scholar
  6. Joseph K. Blitzstein and Persi Diaconis. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Mathematics, 6(4):489-522, 2011. Google Scholar
  7. D. Burstein and J. Rubin. Sufficient conditions for graphicality of bidegree sequences. SIAM J. Discr. Math., 31:50-62, 2017. Google Scholar
  8. David Burstein and Jonathan Rubin. Sufficient conditions for graphicality of bidegree sequences. SIAM Journal on Discrete Mathematics, 31(1):50-62, 2017. Google Scholar
  9. A. A. Chernyak, Z. A. Chernyak, and R. I. Tyshkevich. On forcibly hereditary p-graphical sequences. Discr. Math., 64:111-128, 1987. Google Scholar
  10. Sheshayya A. Choudum. A simple proof of the Erdös-Gallai theorem on graph sequences. Bull. Austral. Math. Soc., 33(1):67-70, 1991. Google Scholar
  11. V Chungphaisan. Conditions for sequences to be r-graphic. Discr. Math., 7(1-2):31-39, 1974. Google Scholar
  12. Brian Cloteaux. Fast sequential creation of random realizations of degree sequences. Internet Mathematics, 12(3):205-219, 2016. Google Scholar
  13. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2009. Google Scholar
  14. Geir Dahl and Truls Flatberg. A remark concerning graphical sequences. Discr. Math., 304(1-3):62-64, 2005. Google Scholar
  15. Dóra Erdös, Rainer Gemulla, and Evimaria Terzi. Reconstructing graphs from neighborhood data. ACM Trans. Knowledge Discovery from Data, 8(4):23:1-23:22, 2014. Google Scholar
  16. Paul Erdös and Tibor Gallai. Graphs with prescribed degrees of vertices [hungarian]. Matematikai Lapok, 11:264-274, 1960. Google Scholar
  17. D. Gale. A theorem on flows in networks. Pacific J. Math., 7:1073-1082, 1957. Google Scholar
  18. Gautam Gupta, Puneet Joshi, and Amitabha Tripathi. Graphic sequences of trees and a problem of Frobenius. Czechoslovak Math. J., 57:49-52, 2007. Google Scholar
  19. S. Louis Hakimi. On realizability of a set of integers as degrees of the vertices of a linear graph -I. SIAM J. Appl. Math., 10(3):496-506, 1962. Google Scholar
  20. Peter L. Hammer, Toshihide Ibaraki, and Bruno Simeone. Threshold sequences. SIAM J. Algebra. Discr., 2(1):39-49, 1981. Google Scholar
  21. Peter L. Hammer and Bruno Simeone. The splittance of a graph. Combinatorica, 1:275-284, 1981. Google Scholar
  22. V. Havel. A remark on the existence of finite graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955. Google Scholar
  23. Heather Hulett, Todd G Will, and Gerhard J Woeginger. Multigraph realizations of degree sequences: Maximization is easy, minimization is hard. Oper. Res. Lett., 36(5):594-596, 2008. Google Scholar
  24. P.J. Kelly. A congruence theorem for trees. Pacific J. Math., 7:961-968, 1957. Google Scholar
  25. Daniel J Kleitman. Minimal number of multiple edges in realization of an incidence sequence without loops. SIAM J. on Applied Math., 18(1):25-28, 1970. Google Scholar
  26. Daniel J Kleitman and Da-Lun Wang. Algorithms for constructing graphs and digraphs with given valences and factors. Discrete Mathematics, 6(1):79-88, 1973. Google Scholar
  27. Manfred Krause. A simple proof of the gale-ryser theorem. The American Mathematical Monthly, 103(4):335-337, 1996. Google Scholar
  28. P. Marchioro, A. Morgana, R. Petreschi, and B. Simeone. Degree sequences of matrogenic graphs. Discrete Mathematics, 51(1):47-61, 1984. Google Scholar
  29. Milena Mihail and Nisheeth Vishnoi. On generating graphs with prescribed degree sequences for complex network modeling applications. 3rd ARACNE, 2002. Google Scholar
  30. Jeffrey W Miller. Reduced criteria for degree sequences. Discrete Mathematics, 313(4):550-562, 2013. Google Scholar
  31. AB Owens and HM Trent. On determining minimal singularities for the realizations of an incidence sequence. SIAM J. on Applied Math., 15(2):406-418, 1967. Google Scholar
  32. Alvin B Owens. On determining the minimum number of multiple edges for an incidence sequence. SIAM J. on Applied Math., 18(1):238-240, 1970. Google Scholar
  33. S. B. Rao. A survey of the theory of potentially p-graphic and forcibly p-graphic degree sequences. In Combinatorics and graph theory, volume 885 of LNM, pages 417-440, 1981. Google Scholar
  34. H.J. Ryser. Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9:371-377, 1957. Google Scholar
  35. E. F. Schmeichel and S. L. Hakimi. On planar graphical degree sequences. SIAM J. Applied Math., 32:598-609, 1977. Google Scholar
  36. Gerard Sierksma and Han Hoogeveen. Seven criteria for integer sequences being graphic. J. Graph Theory, 15(2):223-231, 1991. Google Scholar
  37. Akutsu Tatsuya and Hiroshi Nagamochi. Comparison and enumeration of chemical graphs. Computational and structural biotechnology, 5, 2013. Google Scholar
  38. Amitabha Tripathi and Himanshu Tyagi. A simple criterion on degree sequences of graphs. Discr. Appl. Math., 156(18):3513-3517, 2008. Google Scholar
  39. Amitabha Tripathi, Sushmita Venugopalan, and Douglas B. West. A short constructive proof of the Erdös-Gallai characterization of graphic lists. Discr. Math., 310(4):843-844, 2010. Google Scholar
  40. Amitabha Tripathi and Sujith Vijay. A note on a theorem of Erdös & Gallai. Discr. Math., 265(1-3):417-420, 2003. Google Scholar
  41. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: a survey, I. Cybernetics, 23:734-745, 1987. Google Scholar
  42. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: a survey, II. Cybernetics, 24:137-152, 1988. Google Scholar
  43. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: a survey, III. Cybernetics, 24:539-548, 1988. Google Scholar
  44. Regina Tyshkevich. Decomposition of graphical sequences and unigraphs. Discr. Math., 220:201-238, 2000. Google Scholar
  45. S.M. Ulam. A collection of mathematical problems. Wiley, 1960. Google Scholar
  46. N.C. Wormald. Models of random regular graphs. Surveys in Combin., 267:239-298, 1999. Google Scholar
  47. Igor E Zverovich and Vadim E Zverovich. Contributions to the theory of graphic sequences. Discrete Mathematics, 105(1-3):293-303, 1992. Google Scholar
  48. Igor E. Zverovich and Vadim E. Zverovich. Contributions to the theory of graphic sequences. Discr. Math., 105(1-3):293-303, 1992. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail