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Canonization of a Random Circulant Graph by Counting Walks

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WALCOM: Algorithms and Computation (WALCOM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14549))

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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.

O. Verbitsky was supported by DFG grant KO 1053/8–2. He is on leave from the IAPMM, Lviv, Ukraine.

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Notes

  1. 1.

    Recall that \(\mathbb {Q}(\zeta _n)\) is obtained by adjoining \(\zeta _n\) to the field of rationals \(\mathbb {Q}\). In other words, this is the smallest subfield of \(\mathbb {C}\) containing \(\mathbb {Q}\) and \(\zeta _n\).

References

  1. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1998)

    Google Scholar 

  2. Babai, L.: Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hung. 29(3–4), 329–336 (1977). https://doi.org/10.1007/BF01895854

    Article  MathSciNet  Google Scholar 

  3. Babai, L., Chen, X., Sun, X., Teng, S., Wilmes, J.: Faster canonical forms for strongly regular graphs. In: 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pp. 157–166. IEEE Computer Society (2013). https://doi.org/10.1109/FOCS.2013.25

  4. Babai, L., Erdős, P., Selkow, S.M.: Random graph isomorphism. SIAM J. Comput. 9(3), 628–635 (1980)

    Article  MathSciNet  Google Scholar 

  5. Bach, E., Shallit, J.: Algorithmic Number Theory: Efficient Algorithms. The MIT Press, Cambridge (1996)

    Google Scholar 

  6. Berkholz, C., Bonsma, P.S., Grohe, M.: Tight lower and upper bounds for the complexity of canonical colour refinement. Theory Comput. Syst. 60(4), 581–614 (2017). https://doi.org/10.1007/s00224-016-9686-0

    Article  MathSciNet  Google Scholar 

  7. Bhoumik, S., Dobson, T., Morris, J.: On the automorphism groups of almost all circulant graphs and digraphs. Ars Math. Contemp. 7(2), 499–518 (2014). https://doi.org/10.26493/1855-3974.315.868

    Article  MathSciNet  Google Scholar 

  8. Bollobás, B.: Distinguishing vertices of random graphs. Ann. Discrete Math. 13, 33–49 (1982). https://doi.org/10.1016/S0304-0208(08)73545-X

    Article  MathSciNet  Google Scholar 

  9. Bose, A., Saha, K.: Random Circulant Matrices. CRC Press, Boca Raton (2019). https://doi.org/10.1201/9780429435508

    Book  Google Scholar 

  10. Cai, J., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identifications. Combinatorica 12(4), 389–410 (1992). https://doi.org/10.1007/BF01305232

    Article  MathSciNet  Google Scholar 

  11. Cardon, A., Crochemore, M.: Partitioning a graph in \(O(|A| \log _2 |V|)\). Theor. Comput. Sci. 19, 85–98 (1982)

    Article  Google Scholar 

  12. Davis, P.J.: Circulant Matrices, 2nd edn. AMS Chelsea Publishing, New York (1994)

    Google Scholar 

  13. Dobson, E., Spiga, P., Verret, G.: Cayley graphs on abelian groups. Combinatorica 36(4), 371–393 (2016). https://doi.org/10.1007/s00493-015-3136-5

    Article  MathSciNet  Google Scholar 

  14. Dobson, T., Malnič, A., Marušič, D.: Symmetry in Graphs. Cambridge Studies in Advanced Mathematics, vol. 198. Cambridge University Press, Cambridge (2022). https://doi.org/10.1017/9781108553995

    Book  Google Scholar 

  15. Drozd, Y.A., Kirichenko, V.V.: Finite Dimensional Algebras. Springer, Berlin (1994). https://doi.org/10.1007/978-3-642-76244-4

    Book  Google Scholar 

  16. Evdokimov, S., Ponomarenko, I.: Circulant graphs: recognizing and isomorphism testing in polynomial time. St. Petersburg Math. J. 15(6), 813–835 (2004)

    Article  MathSciNet  Google Scholar 

  17. Fuhlbrück, F., Köbler, J., Ponomarenko, I., Verbitsky, O.: The Weisfeiler-Leman algorithm and recognition of graph properties. Theor. Comput. Sci. 895, 96–114 (2021). https://doi.org/10.1016/j.tcs.2021.09.033

    Article  MathSciNet  Google Scholar 

  18. Godsil, C.: Controllable subsets in graphs. Ann. Comb. 16(4), 733–744 (2012). https://doi.org/10.1007/s00026-012-0156-3

    Article  MathSciNet  Google Scholar 

  19. Hagos, E.M.: Some results on graph spectra. Linear Algebra Appl. 356(1–3), 103–111 (2002). https://doi.org/10.1016/S0024-3795(02)00324-5

    Article  MathSciNet  Google Scholar 

  20. Immerman, N., Lander, E.: Describing graphs: a first-order approach to graph canonization. In: Selman, A.L. (ed.) Complexity Theory Retrospective, pp. 59–81. Springer, New York (1990). https://doi.org/10.1007/978-1-4612-4478-3_5

    Chapter  Google Scholar 

  21. Kiefer, S., Ponomarenko, I., Schweitzer, P.: The Weisfeiler-Leman dimension of planar graphs is at most 3. J. ACM 66(6), 44:1–44:31 (2019). https://doi.org/10.1145/3333003

  22. Kluge, L.: Combinatorial refinement on circulant graphs. Technical report (2022). arxiv.org/abs/2204.01054

  23. Kucera, L.: Canonical labeling of regular graphs in linear average time. In: 28th Annual Symposium on Foundations of Computer Science (FOCS 1987), pp. 271–279 (1987). https://doi.org/10.1109/SFCS.1987.11

  24. Liu, F., Siemons, J.: Unlocking the walk matrix of a graph. J. Algebraic Comb. 55(3), 663–690 (2022). https://doi.org/10.1007/s10801-021-01065-3

    Article  MathSciNet  Google Scholar 

  25. McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014). https://doi.org/10.1016/j.jsc.2013.09.003

    Article  MathSciNet  Google Scholar 

  26. Meckes, M.W.: Some results on random circulant matrices. In: High Dimensional Probability. V: The Luminy Volume, pp. 213–223. IMS, Institute of Mathematical Statistics, Beachwood (2009). https://doi.org/10.1214/09-IMSCOLL514

  27. Muzychuk, M.: A solution of the isomorphism problem for circulant graphs. Proc. Lond. Math. Soc. 88(1), 1–41 (2004). https://doi.org/10.1112/S0024611503014412

    Article  MathSciNet  Google Scholar 

  28. Muzychuk, M.E., Klin, M.H., Pöschel, R.: The isomorphism problem for circulant graphs via Schur ring theory. In: Codes and Association Schemes. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 56, pp. 241–264. DIMACS/AMS (1999). https://doi.org/10.1090/dimacs/056/19

  29. O’Rourke, S., Touri, B.: On a conjecture of Godsil concerning controllable random graphs. SIAM J. Control. Optim. 54(6), 3347–3378 (2016). https://doi.org/10.1137/15M1049622

    Article  MathSciNet  Google Scholar 

  30. Ponomarenko, I.: On the WL-dimension of circulant graphs of prime power order. Technical report (2022). http://arxiv.org/abs/2206.15028

  31. Powers, D.L., Sulaiman, M.M.: The walk partition and colorations of a graph. Linear Algebra Appl. 48, 145–159 (1982). https://doi.org/10.1016/0024-3795(82)90104-5

    Article  MathSciNet  Google Scholar 

  32. Verbitsky, O., Zhukovskii, M.: Canonization of a random circulant graph by counting walks. Technical report (2023). arxiv.org/abs/2310.05788

  33. Weisfeiler, B., Leman, A.: 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

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Authors and Affiliations

  1. Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany

    Oleg Verbitsky

  2. Department of Computer Science, University of Sheffield, Sheffield, UK

    Maksim Zhukovskii

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  1. Oleg Verbitsky
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  2. Maksim Zhukovskii
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Correspondence to Oleg Verbitsky .

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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Ryuhei Uehara

  2. Iwate University, Morioka, Japan

    Katsuhisa Yamanaka

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

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Verbitsky, O., Zhukovskii, M. (2024). Canonization of a Random Circulant Graph by Counting Walks. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_23

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  • DOI: https://doi.org/10.1007/978-981-97-0566-5_23

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