skip to main content
research-article

On Minrank and Forbidden Subgraphs

Authors Info & Claims
Published:07 May 2019Publication History
Skip Abstract Section

Abstract

The minrank over a field F of a graph G on the vertex set { 1,2,… ,n} is the minimum possible rank of a matrix M ∈ Fn × n such that Mi, i ≠ 0 for every i, and Mi, j =0 for every distinct non-adjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n,H, F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this article, we study this quantity for various graphs H and fields F. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F), which yields a nearly tight bound of Ω (√ n/ log n) for the triangle H=K3. For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n,H, R) ≥ nδ for some δ = δ (H)> 0. As a by-product of this construction, we disprove a conjecture of Codenotti et al. [11]. The results are motivated by questions in information theory, circuit complexity, and geometry.

References

  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. 1980. A note on Ramsey numbers. J. Comb. Theory, Ser. A 29, 3 (1980), 354--360.Google ScholarGoogle ScholarCross RefCross Ref
  2. Noga Alon. 1994. Explicit Ramsey graphs and orthonormal labelings. Electr. J. Comb. 1, R12 (1994).Google ScholarGoogle Scholar
  3. Noga Alon. 2002. Graph powers. In Contemporary Combinatorics, B. Bollobás (Ed.). Springer, 11--28.Google ScholarGoogle Scholar
  4. Noga Alon, László Babai, and H. Suzuki. 1991. Multilinear polynomials and Frankl--Ray-Chaudhuri--Wilson type intersection theorems. J. Comb. Theory, Ser. A 58, 2 (1991), 165--180. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Noga Alon, Igor Balla, Lior Gishboliner, Adva Mond, and Frank Mousset. 2018. The minrank of random graphs over arbitrary fields. Arxiv abs/1809.01873 (2018).Google ScholarGoogle Scholar
  6. Noga Alon and Joel H. Spencer. 2016. The Probabilistic Method (4th ed.). Wiley Publishing. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Ziv Bar-Yossef, Yitzhak Birk, T. S. Jayram, and Tomer Kol. 2006. Index coding with side information. In FOCS. 197--206. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Anna Blasiak, Robert Kleinberg, and Eyal Lubetzky. 2013. Broadcasting with side information: Bounding and approximating the broadcast rate. IEEE Trans. Inform. Theory 59, 9 (2013), 5811--5823. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Yair Caro, Yusheng Li, Cecil C. Rousseau, and Yuming Zhang. 2000. Asymptotic bounds for some bipartite graph: Complete graph Ramsey numbers. Discrete Mathematics 220, 1--3 (2000), 51--56. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Eden Chlamtáč and Ishay Haviv. 2014. Linear index coding via semidefinite programming. Combinatorics, Probability 8 Computing 23, 2 (2014), 223--247. Preliminary version in SODA’12.Google ScholarGoogle Scholar
  11. Bruno Codenotti, Pavel Pudlák, and Giovanni Resta. 2000. Some structural properties of low-rank matrices related to computational complexity. Theor. Comput. Sci. 235, 1 (2000), 89--107. Preliminary version in ECCC’97. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Reinhard Diestel. 2017. Graph Theory (5th ed.). Graduate texts in mathematics, Vol. 173. Springer-Verlag, Berlin.. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Paul Erdös, Ralph J. Faudree, Cecil C. Rousseau, and Richard H. Schelp. 1978. On cycle-complete graph Ramsey numbers. J. Graph Theory 2, 1 (1978), 53--64.Google ScholarGoogle ScholarCross RefCross Ref
  14. Paul Erdös and László Lovász. 1975. Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and Finite Sets, A. Hajnal, R. Rado, and V. T. Sós (Eds.). North-Holland, Amsterdam, 609--627.Google ScholarGoogle Scholar
  15. Paul Erdös, Robert J. McEliece, and Herbert Taylor. 1971. Ramsey bounds for graph products. Pacific J. Math. 37, 1 (1971), 45--46.Google ScholarGoogle ScholarCross RefCross Ref
  16. Alexander Golovnev, Oded Regev, and Omri Weinstein. 2017. The minrank of random graphs. In Randomization and Approximation Techniques in Computer Science (RANDOM). 46:1--46:13.Google ScholarGoogle Scholar
  17. Willem Haemers. 1979. On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 2 (1979), 231--232. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Willem Haemers. 1981. An upper bound for the Shannon capacity of a graph. In Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978). Colloq. Math. Soc. János Bolyai, Vol. 25. North-Holland, Amsterdam, 267--272.Google ScholarGoogle Scholar
  19. Ishay Haviv. 2018. On minrank and the Lovász theta function. In International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’18), Vol. 116. 13:1--13:15.Google ScholarGoogle Scholar
  20. Ishay Haviv and Michael Langberg. 2013. H-wise independence. In Innovations in Theoretical Computer Science (ITCS’13). 541--552. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. B. S. Kashin and S. V. Konyagin. 1981. Systems of vectors in Hilbert space. In Number Theory, Mathematical Analysis, and Their Applications. Trudy Mat. Inst. Steklov., Vol. 157. 64--67.Google ScholarGoogle Scholar
  22. S. V. Konyagin. 1981. Systems of vectors in Euclidean space and an extremal problem for polynomials. Mat. Zametki 29, 1 (1981), 63--74.Google ScholarGoogle Scholar
  23. László Lovász. 1979. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 1 (1979), 1--7. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. René Peeters. 1996. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica 16, 3 (1996), 417--431.Google ScholarGoogle ScholarCross RefCross Ref
  25. Pavel Pudlák. 2002. Cycles of nonzero elements in low rank matrices. Combinatorica 22, 2 (2002), 321--334.Google ScholarGoogle ScholarCross RefCross Ref
  26. Pavel Pudlák, Vojtech Rödl, and Jirí Sgall. 1997. Boolean circuits, tensor ranks, and communication complexity. SIAM J. Comput. 26, 3 (1997), 605--633. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Søren Riis. 2007. Information flows, graphs and their guessing numbers. Electr. J. Comb. 14, 1 (2007).Google ScholarGoogle Scholar
  28. Moshe Rosenfeld. 1991. Almost orthogonal lines in E<sup>d</sup>. DIMACS Series in Discrete Math. 4 (1991), 489--492.Google ScholarGoogle ScholarCross RefCross Ref
  29. Claude E. Shannon. 1956. The zero error capacity of a noisy channel. Institute of Radio Engineers, Transactions on Information Theory IT-2 (1956), 8--19.Google ScholarGoogle Scholar
  30. Joel Spencer. 1977. Asymptotic lower bounds for Ramsey functions. Discrete Mathematics 20 (1977), 69--76.Google ScholarGoogle ScholarCross RefCross Ref
  31. Benny Sudakov. 2002. A note on odd cycle-complete graph Ramsey numbers. Electr. J. Comb. 9, 1 (2002).Google ScholarGoogle Scholar
  32. Leslie G. Valiant. 1977. Graph-theoretic arguments in low-level complexity. In Mathematical Foundations of Computer Science (MFCS), 6th Symposium. 162--176.Google ScholarGoogle Scholar
  33. Leslie G. Valiant. 1992. Why is Boolean complexity theory difficult? In Poceedings of the London Mathematical Society Symposium on Boolean Function Complexity, Vol. 169. 84--94. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Xiaodong Xu, Xie Zheng, Geoffrey Exoo, and Stanislaw P. Radziszowski. 2004. Constructive lower bounds on classical multicolor Ramsey numbers. Electr. J. Comb. 11, 1 (2004).Google ScholarGoogle Scholar

Index Terms

  1. On Minrank and Forbidden Subgraphs

          Recommendations

          Comments

          Login options

          Check if you have access through your login credentials or your institution to get full access on this article.

          Sign in

          Full Access

          • Published in

            cover image ACM Transactions on Computation Theory
            ACM Transactions on Computation Theory  Volume 11, Issue 4
            December 2019
            252 pages
            ISSN:1942-3454
            EISSN:1942-3462
            DOI:10.1145/3331049
            Issue’s Table of Contents

            Copyright © 2019 ACM

            Publisher

            Association for Computing Machinery

            New York, NY, United States

            Publication History

            • Published: 7 May 2019
            • Accepted: 1 March 2019
            • Revised: 1 January 2019
            • Received: 1 September 2018
            Published in toct Volume 11, Issue 4

            Permissions

            Request permissions about this article.

            Request Permissions

            Check for updates

            Qualifiers

            • research-article
            • Research
            • Refereed

          PDF Format

          View or Download as a PDF file.

          PDF

          eReader

          View online with eReader.

          eReader

          HTML Format

          View this article in HTML Format .

          View HTML Format
          About Cookies On This Site

          We use cookies to ensure that we give you the best experience on our website.

          Learn more

          Got it!