Ramsey numbers and an approximation algorithm for the vertex cover problem

SummaryWe show two results. First we derive an upper bound for the special Ramsey numbers rk(q) where rk(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).