Summary
We show two results. First we derive an upper bound for the special Ramsey numbers r k(q) where r k(q) is the largest number of nodes a graph without odd cycles of length bounded by 2k+1 and without an independent set of size q+1 can have. We prove \(r_k (q) \leqq \frac{k}{{k + {\text{1}}}}q^{\frac{{k + {\text{1}}}}{k}} + \frac{{k + {\text{2}}}}{{k + {\text{1}}}}q\) The proof is constructive and yields an algorithm for computing an independent set of that size. Using this algorithm we secondly describe an O(¦V¦·¦E¦) time bounded approximation algorithm for the vertex cover problem, whose worst case ratio is \(\Delta \leqq {\text{2 - }}\frac{{\text{1}}}{{k + {\text{1}}}}\), for all graphs with at most (2k+3)k(2k+2) nodes (e.g. Δ≦1.8, if ¦V¦≦146000).
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Ajtai, M., Komlós, J., Szemeredi, E.: A note on Ramsey numbers. J. Comb. Theory, Ser. A 29, 354–360 (1980)
Bar-Yehuda, R., Even, S.: On approximating a vertex cover for planar graphs. Proc. 14th Ann. ACM Symp. Th. Computing, pp. 303–309. San Francisco 1982
Bar-Yehuda, R, Even, S.: A 2−log logn/2logn performance ratio for the weighted vertex cover problem. Technion Haifa, Technical Report #260, January 1983
Erdös, P.: Graph theory and probability, Can. J. Math. 11, 34–38 (1959)
Erdös, P., Gallai, T.: On the minimal number of vertices representing the edges of a graph, pp. 181–203. Magyar Tudományos Akadémia Matematikai Kutato Intézetének Közleményei VI. Evfolyam 1961
Even, S.: Graph algorithms. Rockville, MD: Computer Science Press 1979
Fajtlowicz, S.: On the size of independent sets in graphs. In: Proc. 9th S.E. Conf. Combinatorics, Graph Theory and Computing, Boca Raton, Congressum Numerantium No. XXI, pp. 269–274. Utilitas Mathematical-Publ. Inc. 1978
Graver, J.E, Yackel, J.: Some graph theoretic results with Ramsey's theorem. J. Comb. Theory, Ser. A 4, 125–175 (1968)
Griggs, J.R.: Lower bounds on the independence number in terms of the degrees. J. Comb. Theory, Ser. B 34, 22–39 (1983)
Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Appl. Math. 6, 243–254 (1983)
Hopkins, G., Staton, W.: Girth and independence ratio. Can. Math. Bull. 25, 179–186 (1982)
König, D.: Theorie der endlichen und unendlichen Graphen. Leipzig: Akademische Verlagsgesellschaft 1936
Monien, B.: The complexity of determining a shortest cycle of even length. Computing 31, 355–369 (1983)
Monien, B., Speckenmeyer, E.: Some further approximation algorithms for the vertex cover problem. In: Proc. 8th Coll. on Trees in Algebra and Programming (CAAP 83), L'Aquila. Lecture Notes in Computer Science 159, 341–349 (1983)
Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Math. Program. 8, 232–248 (1975)
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Monien, B., Speckenmeyer, E. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica 22, 115–123 (1985). https://doi.org/10.1007/BF00290149
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DOI: https://doi.org/10.1007/BF00290149