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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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