Narrow Results

Ramsey numbers deal with conditions when a combinatorial object necessarily contains some smaller given objects. It is well known that it is very difficult to obtain the values of Ramsey numbers. In this work, a theoretical chemical/biological solution is presented in terms of membrane computing for the decision version of Ramsey number problem, that is, to decide whether an integer n is the value of Ramsey number R(k, l), where k and l are integers.

An algorithm that produces polygonal cable links is described, and applications discussed. In particular, the stick numbers of T_p,q torus links are shown to be 4p for 2p < q ≤ 3p, and it is shown that, in general, . Further, it is shown that the Ramsey number of a link is at least the sum of its arc index and bridge number. Using these results, we relate the Ramsey, stick and crossing numbers of torus links, showing .

For two given graphs G₁ and G₂, the Ramsey number R(G₁, G₂) is the smallest positive integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let S_n denote a star of order n and W_m a wheel of order m + 1. In this paper we show that R(S_n, W₈) = 2n + 1 for n ≥ 5 and n ≡ 1 (mod 2).

Let H be a graph with the chromatic number h and the chromatic surplus s. A connected graph G of order n is called H-good if R(G, H) = (n - 1)(h - 1) + s. In this paper, we show that P_n is 2K_m-good for n ≥ 3. Furthermore, we obtain the Ramsey number R(L, 2K_m), where L is a linear forest. Moreover, we also give the Ramsey number R(L, H_m) which is an extension for R(kP_n, H_m) proposed by Ali et al. [1], where H_m is a cocktail party graph on 2m vertices.