A limit theorem for the Shannon capacities of odd cycles. II

Bohman, Tom

doi:10.1090/S0002-9939-04-07470-2

A limit theorem for the Shannon capacities of odd cycles. II
HTML articles powered by AMS MathViewer

by Tom Bohman PDF

Proc. Amer. Math. Soc. 133 (2005), 537-543 Request permission

Abstract:

It follows from a construction for independent sets in the powers of odd cycles given in the predecessor of this paper that the limit as $k$ goes to infinity of $k + 1/2 - \Theta ( C_{2k+1} )$ is zero, where $\Theta (G)$ is the Shannon capacity of a graph $G$. This paper contains a shorter proof of this limit theorem that is based on an ‘expansion process’ introduced in an older paper of L. Baumert, R. McEliece, E. Rodemich, H. Rumsey, R. Stanley and H. Taylor. We also refute a conjecture from that paper, using ideas from the predecessor of this paper.

References

N. Alon, Graph powers, Contemporary combinatorics, Bolyai Soc. Math. Stud., vol. 10, János Bolyai Math. Soc., Budapest, 2002, pp. 11–28. MR 1919567
L. D. Baumert, R. J. McEliece, Eugene Rodemich, Howard C. Rumsey Jr., Richard Stanley, and Herbert Taylor, A combinatorial packing problem, Computers in algebra and number theory (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970) SIAM-AMS Proc., Vol. IV, Amer. Math. Soc., Providence, R.I., 1971, pp. 97–108. MR 0337668
Claude Berge, Motivations and history of some of my conjectures, Discrete Math. 165/166 (1997), 61–70. Graphs and combinatorics (Marseille, 1995). MR 1439260, DOI 10.1016/S0012-365X(96)00161-6
T. Bohman, A limit theorem for the Shannon capacities of odd cycles I, Proceedings of the AMS 131 (2003), 3559–3569.
Tom Bohman and Ron Holzman, A nontrivial lower bound on the Shannon capacities of the complements of odd cycles, IEEE Trans. Inform. Theory 49 (2003), no. 3, 721–722. MR 1967195, DOI 10.1109/TIT.2002.808128
T. Bohman, M. Ruszinkó, L. Thoma, Shannon capacity of large odd cycles, Proceedings of the 2000 IEEE International Symposium on Information Theory, June 25-30, Sorrento, Italy, p. 179.
M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The Strong Perfect Graph Theorem, submitted.
R. S. Hales, Numerical invariants and the strong product of graphs, J. Combinatorial Theory Ser. B 15 (1973), 146–155. MR 321810, DOI 10.1016/0095-8956(73)90014-2
János Körner and Alon Orlitsky, Zero-error information theory, IEEE Trans. Inform. Theory 44 (1998), no. 6, 2207–2229. Information theory: 1948–1998. MR 1658803, DOI 10.1109/18.720537
László Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), no. 1, 1–7. MR 514926, DOI 10.1109/TIT.1979.1055985
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335