A limit theorem for the Shannon capacities of odd cycles. II
Author:
Tom Bohman
Journal:
Proc. Amer. Math. Soc. 133 (2005), 537543
MSC (2000):
Primary 94A15, 05C35, 05C38
Published electronically:
September 8, 2004
MathSciNet review:
2093078
Fulltext PDF Free Access
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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 goes to infinity of is zero, where is the Shannon capacity of a graph . 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.
 1.
N.
Alon, Graph powers, Contemporary combinatorics, Bolyai Soc.
Math. Stud., vol. 10, János Bolyai Math. Soc., Budapest, 2002,
pp. 11–28. MR 1919567
(2003h:05181)
 2.
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. SIAMAMS Sympos. Appl. Math., New York, 1970)
Amer. Math. Soc., Providence, R.I., 1971, pp. 97–108. SIAMAMS
Proc., Vol. IV. MR 0337668
(49 #2437)
 3.
Claude
Berge, Motivations and history of some of my conjectures,
Discrete Math. 165/166 (1997), 61–70. Graphs and
combinatorics (Marseille, 1995). MR 1439260
(98a:05091), http://dx.doi.org/10.1016/S0012365X(96)001616
 4.
T. Bohman, A limit theorem for the Shannon capacities of odd cycles I, Proceedings of the AMS 131 (2003), 35593569.
 5.
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
(2004b:94039), http://dx.doi.org/10.1109/TIT.2002.808128
 6.
T. Bohman, M. Ruszinkó, L. Thoma, Shannon capacity of large odd cycles, Proceedings of the 2000 IEEE International Symposium on Information Theory, June 2530, Sorrento, Italy, p. 179.
 7.
M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The Strong Perfect Graph Theorem, submitted.
 8.
R.
S. Hales, Numerical invariants and the strong product of
graphs, J. Combinatorial Theory Ser. B 15 (1973),
146–155. MR 0321810
(48 #177)
 9.
János
Körner and Alon
Orlitsky, Zeroerror information theory, IEEE Trans. Inform.
Theory 44 (1998), no. 6, 2207–2229. Information
theory: 1948–1998. MR 1658803
(99h:94034), http://dx.doi.org/10.1109/18.720537
 10.
László
Lovász, On the Shannon capacity of a graph, IEEE Trans.
Inform. Theory 25 (1979), no. 1, 1–7. MR 514926
(81g:05095), http://dx.doi.org/10.1109/TIT.1979.1055985
 11.
Claude
E. Shannon, The zero error capacity of a noisy channel,
Institute of Radio Engineers, Transactions on Information Theory,
IT2 (1956), no. September, 8–19. MR 0089131
(19,623b)
 1.
 N. Alon, Graph Powers, Contemporary Combinatorics (B. Bollobás, ed.), Bolyai Society Mathematical Studies, Springer 2002, pp. 1128. MR 2003h:05181
 2.
 L. Baumert, R. McEliece, E. Rodemich, H. Rumsey, R. Stanley, H. Taylor, A Combinatorial Packing Problem, Computers in Algebra and Number Theory, SIAMAMS Proc., vol. 4, Providence, American Mathematical Society; 1971, pp. 97108. MR 49:2437
 3.
 C. Berge, Motivations and history of some of my conjectures Discrete Mathematics 165 (1997), 6170. MR 98a:05091
 4.
 T. Bohman, A limit theorem for the Shannon capacities of odd cycles I, Proceedings of the AMS 131 (2003), 35593569.
 5.
 T. Bohman, R. Holzman, A nontrivial lower bound on the Shannon capacities of the complements of odd cycles, IEEE Transactions on Information Theory, 49(3) (2003), 721722. MR 2004b:94039
 6.
 T. Bohman, M. Ruszinkó, L. Thoma, Shannon capacity of large odd cycles, Proceedings of the 2000 IEEE International Symposium on Information Theory, June 2530, Sorrento, Italy, p. 179.
 7.
 M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The Strong Perfect Graph Theorem, submitted.
 8.
 R. S. Hales, Numerical invariants and the strong product of graphs, Journal of Combinatorial Theory  B, 15 (1973), 146155. MR 48:177
 9.
 J. Körner and A. Orlitsky, Zeroerror information theory, IEEE Transactions on Information Theory 44(6) (1998), 22072229. MR 99h:94034
 10.
 L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory 25(1) (1979), 17. MR 81g:05095
 11.
 C. E. Shannon, The zeroerror capacity of a noisy channel, IRE Transactions on Information Theory, 2(3) (1956), 819. MR 19:623b
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Additional Information
Tom Bohman
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
tbohman@moser.math.cmu.edu
DOI:
http://dx.doi.org/10.1090/S0002993904074702
PII:
S 00029939(04)074702
Keywords:
Shannon capacity,
odd cycles
Received by editor(s):
May 30, 2003
Received by editor(s) in revised form:
August 5, 2003
Published electronically:
September 8, 2004
Additional Notes:
This research was supported in part by NSF Grant DMS0100400.
Communicated by:
John R. Stembridge
Article copyright:
© Copyright 2004
American Mathematical Society
