Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
156 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Canonization of a random circulant graph by counting walks (2310.05788v2)

Published 9 Oct 2023 in cs.CC

Abstract: It is well known that almost all graphs are canonizable by a simple combinatorial routine known as color refinement. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of color refinement with vertex individualization produces a canonical labeling for almost all circulant digraphs (Cayley digraphs of a cyclic group). To our best knowledge, this is the first application of combinatorial refinement in the realm of vertex-transitive graphs. Remarkably, we do not even need the full power of the color refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are canonizable by Tinhofer's canonization procedure. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the 2-dimensional Weisfeiler-Leman algorithm.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (31)
  1. T. M. Apostol. Introduction to analytic number theory. New York, NY: Springer, 1998.
  2. L. Babai. Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hungar., 29(3-4):329–336, 1977.
  3. Faster canonical forms for strongly regular graphs. In 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS’13), pages 157–166. IEEE Computer Society, 2013.
  4. Random graph isomorphism. SIAM Journal on Computing, 9(3):628–635, 1980.
  5. E. Bach and J. Shallit. Algorithmic number theory, Vol. 1: Efficient algorithms. Cambridge, MA: The MIT Press, 1996.
  6. Tight lower and upper bounds for the complexity of canonical colour refinement. Theory Comput. Syst., 60(4):581–614, 2017.
  7. On the automorphism groups of almost all circulant graphs and digraphs. Ars Math. Contemp., 7(2):499–518, 2014.
  8. B. Bollobás. Distinguishing vertices of random graphs. Annals of Discrete Mathematics, 13:33–49, 1982.
  9. A. Bose and K. Saha. Random circulant matrices. Boca Raton, FL: CRC Press, 2019.
  10. An optimal lower bound on the number of variables for graph identifications. Combinatorica, 12(4):389–410, 1992.
  11. A. Cardon and M. Crochemore. Partitioning a graph in O⁢(|A|⁢log2⁡|V|)𝑂𝐴subscript2𝑉O(|A|\log_{2}|V|)italic_O ( | italic_A | roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_V | ). Theor. Comput. Sci., 19:85–98, 1982.
  12. P. J. Davis. Circulant matrices. New York, NY: AMS Chelsea Publishing, 2nd ed. edition, 1994.
  13. Cayley graphs on abelian groups. Combinatorica, 36(4):371–393, 2016.
  14. Symmetry in graphs, volume 198 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2022.
  15. Finite dimensional algebras. Springer-Verlag, Berlin, 1994.
  16. S. Evdokimov and I. Ponomarenko. Circulant graphs: recognizing and isomorphism testing in polynomial time. St. Petersbg. Math. J., 15(6):813–835, 2004.
  17. The Weisfeiler-Leman algorithm and recognition of graph properties. Theor. Comput. Sci., 895:96–114, 2021.
  18. C. Godsil. Controllable subsets in graphs. Ann. Comb., 16(4):733–744, 2012.
  19. E. M. Hagos. Some results on graph spectra. Linear Algebra Appl., 356(1-3):103–111, 2002.
  20. N. Immerman and E. Lander. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective, pages 59–81. Springer, 1990.
  21. The Weisfeiler-Leman dimension of planar graphs is at most 3. J. ACM, 66(6):44:1–44:31, 2019.
  22. L. Kucera. Canonical labeling of regular graphs in linear average time. In 28th Annual Symposium on Foundations of Computer Science (FOCS’87), pages 271–279, 1987.
  23. F. Liu and J. Siemons. Unlocking the walk matrix of a graph. J. Algebr. Comb., 55(3):663–690, 2022.
  24. B. D. McKay and A. Piperno. Practical graph isomorphism, ii. Journal of Symbolic Computation, 60:94–112, 2014.
  25. M. W. Meckes. Some results on random circulant matrices. In High dimensional probability. V: The Luminy volume., pages 213–223. Beachwood, OH: IMS, Institute of Mathematical Statistics, 2009.
  26. M. Muzychuk. A solution of the isomorphism problem for circulant graphs. Proc. Lond. Math. Soc. (3), 88(1):1–41, 2004.
  27. The isomorphism problem for circulant graphs via Schur ring theory. In Codes and Association Schemes, volume 56 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 241–264. DIMACS/AMS, 1999.
  28. S. O’Rourke and B. Touri. On a conjecture of Godsil concerning controllable random graphs. SIAM J. Control. Optim., 54(6):3347–3378, 2016.
  29. I. Ponomarenko. On the WL-dimension of circulant graphs of prime power order. E-print, http://arxiv.org/abs/2206.15028, 2022.
  30. The walk partition and colorations of a graph. Linear Algebra Appl., 48:145–159, 1982.
  31. B. Weisfeiler and A. Leman. The reduction of a graph to canonical form and the algebra which appears therein. NTI, Ser. 2, 9:12–16, 1968. English translation is available at https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets