Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
167 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 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

On the Ratio of Shannon Numbers of Graphs (2307.06155v5)

Published 12 Jul 2023 in math.CO, cs.IT, and math.IT

Abstract: Let $\Gamma$ be a function that maps two arbitrary graphs $G$ and $H$ to a non-negative real number such that $$\alpha(G{\boxtimes n})\leq \alpha(H{\boxtimes n})\Gamma(G,H)n$$ where $n$ is any natural number and $G{\boxtimes n}$ is the strong product of $G$ with itself $n$ times. We establish the equivalence of two different approaches for finding such a function $\Gamma$. The common solution obtained through either approach is termed ``the relative fractional independence number of a graph $G$ with respect to another graph $H$". We show this function by $\alpha*(G|H)$ and discuss some of its properties. In particular, we show that $\alpha*(G|H)\geq \frac{X(G)}{X(H)} \geq \frac{1}{\alpha*(H|G)},$ where $X(G)$ can be the independence number, the Shannon capacity, the fractional independence number, the Lov\'{a}sz number, or the Schrijver's or Szegedy's variants of the Lov\'{a}sz number of a graph $G$. This inequality is the first explicit non-trivial upper bound on the ratio of the invariants of two arbitrary graphs, as mentioned earlier, which can also be used to obtain upper or lower bounds for these invariants. As explicit applications, we present new upper bounds for the ratio of the Shannon capacity of two Cayley graphs and compute new lower bounds on the Shannon capacity of certain Johnson graphs (yielding the exact value of their Haemers number). Moreover, we show that $\alpha*(G|H)$ can be used to present a stronger version of the well-known No-Homomorphism Lemma.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (37)
  1. R. M. Karp, “Reducibility among combinatorial problems,” in Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA (R. E. Miller and J. W. Thatcher, eds.), The IBM Research Symposia Series, pp. 85–103, Plenum Press, New York, 1972.
  2. C. E. Shannon, “The zero error capacity of a noisy channel,” IRE Trans. Inf. Theory, vol. 2, no. 3, pp. 8–19, 1956.
  3. T. Bohman, “A limit theorem for the shannon capacities of odd cycles i,” Proceedings of the American Mathematical Society, vol. 131, no. 11, pp. 3559–3569, 2003.
  4. K. A. Mathew and P. R. Östergård, “New lower bounds for the shannon capacity of odd cycles,” Designs, Codes and Cryptography, vol. 84, no. 1, pp. 13–22, 2017.
  5. I. Sason, “Observations on the lovász θ𝜃\thetaitalic_θ-function, graph capacity, eigenvalues, and strong products,” Entropy, vol. 25, no. 1, p. 104, 2023.
  6. N. Alon and E. Lubetzky, “The shannon capacity of a graph and the independence numbers of its powers,” IEEE Transactions on Information Theory, vol. 52, no. 5, pp. 2172–2176, 2006.
  7. R. Hales, “Numerical invariants and the strong product of graphs,” Journal of Combinatorial Theory, Series B, vol. 15, no. 2, pp. 146–155, 1973.
  8. L. Lovász, “On the shannon capacity of a graph,” IEEE Transactions on Information Theory, vol. 25, no. 1, pp. 1–7, 1979.
  9. W. Haemers, “An upper bound for the shannon capacity of a graph,” in Colloq. Math. Soc. János Bolyai, vol. 25, pp. 267–272, Hungary, 1978.
  10. M. Grötschel, L. Lovász, and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,” Combinatorica, vol. 1, no. 2, pp. 169–197, 1981.
  11. S. Hu, I. Tamo, and O. Shayevitz, “A bound on the shannon capacity via a linear programming variation,” SIAM Journal on Discrete Mathematics, vol. 32, no. 3, pp. 2229–2241, 2018.
  12. B. Bukh and C. Cox, “On a fractional version of haemers’ bound,” IEEE Transactions on Information Theory, vol. 65, no. 6, pp. 3340–3348, 2018.
  13. M. Riddle, Sandwich theorem and calculation of the theta function for several graphs. Brigham Young University, 2003.
  14. N. Alon, “Graph powers, contemporary combinatorics (b. bollobás, ed.), bolyai society mathematical studies,” 2002.
  15. M. Jurkiewicz, “A survey on known values and bounds on the shannon capacity,” Selected Topics in Modern Mathematics; Gancarzewicz, G., Skrzy nski, M., Eds, pp. 115–128, 2014.
  16. J. Korner and A. Orlitsky, “Zero-error information theory,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2207–2229, 1998.
  17. W.-K. Chiang, R.-J. Chen, et al., “The (n, k)-star graph-a generalized star graph,” Information Processing Letters, vol. 56, no. 5, pp. 259–264, 1995.
  18. E. Sonnemann and O. Krafft, “Independence numbers of product graphs,” Journal of Combinatorial Theory, Series B, vol. 17, no. 2, pp. 133–142, 1974.
  19. M. Rosenfeld, “On a problem of ce shannon in graph theory,” Proceedings of the American Mathematical Society, vol. 18, no. 2, pp. 315–319, 1967.
  20. A. Schrijver, “A comparison of the delsarte and lovász bounds,” IEEE Transactions on Information Theory, vol. 25, no. 4, pp. 425–429, 1979.
  21. R. J. McEliece, E. R. Rodemich, and H. C. Rumsey Jr, “The lovász bound and some generalizations,” J. Combin. Inform. System Sci, vol. 3, no. 3, pp. 134–152, 1978.
  22. M. Szegedy, “A note on the ϑitalic-ϑ\varthetaitalic_ϑ number of Lovász and the generalized Delsarte bound,” in Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 36–39, 1994.
  23. W. Haemers et al., “On some problems of lovász concerning the shannon capacity of a graph,” IEEE Transactions on Information Theory, vol. 25, no. 2, pp. 231–232, 1979.
  24. L. Esperet and D. R. Wood, “Colouring strong products,” arXiv preprint arXiv:2205.04953, 2022.
  25. A. Vesel, “The independence number of the strong product of cycles,” Computers and Mathematics with Applications, vol. 36, no. 7, pp. 9–21, 1998.
  26. S. H. Badalyan and S. E. Markosyan, “On the independence number of the strong product of cycle-powers,” Discrete Mathematics, vol. 313, no. 1, pp. 105–110, 2013.
  27. K. Vesztergombi, “Some remarks on the chromatic number of the strong product of graphs,” Acta Cybernetica, vol. 4, no. 2, pp. 207–212, 1979.
  28. M. Jurkiewicz, M. Kubale, and K. Ocetkiewicz, “On the independence number of some strong products of cycle-powers,” Foundations of Computing and Decision Sciences, vol. 40, pp. 133–141, 2015.
  29. S. C. Polak and A. Schrijver, “New lower bound on the shannon capacity of c7subscript𝑐7c_{7}italic_c start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT from circular graphs,” Information Processing Letters, vol. 143, pp. 37–40, 2019.
  30. L. Baumert, R. McEliece, E. Rodemich, H. Rumsey, R. Stanley, and H. Taylor, “A combinatorial packing problem,” Computers in algebra and number theory, vol. 4, 1971.
  31. T. Bohman, R. Holzman, and V. Natarajan, “On the independence numbers of the cubes of odd cycles,” the electronic journal of combinatorics, pp. P10–P10, 2013.
  32. T. Bohman, “A limit theorem for the shannon capacities of odd cycles. ii,” Proceedings of the American Mathematical Society, vol. 133, no. 2, pp. 537–543, 2005.
  33. CRC press, 2011.
  34. M. O. Albertson and K. L. Collins, “Homomorphisms of 3-chromatic graphs,” Discret. Math., vol. 54, no. 2, pp. 127–132, 1985.
  35. M. O. Albertson, L. Chan, and R. Haas, “Independence and graph homomorphisms graph homomorphisms,” Journal of graph theory, vol. 17, no. 5, pp. 581–588, 1993.
  36. A. Acín, R. Duan, D. E. Roberson, A. B. Sainz, and A. Winter, “A new property of the lovász number and duality relations between graph parameters,” Discrete Applied Mathematics, vol. 216, pp. 489–501, 2017.
  37. B. Codenotti, I. Gerace, and G. Resta, “Some remarks on the shannon capacity of odd cycles,” Ars Combinatoria, vol. 66, pp. 243–258, 2003.
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