A neural network parallel algorithm for Ramsey numbers

Abstract

A neural network parallel algorithm for finding Ramsey numbers in two-coloring is described. The Ramsey number R(r,b) is given by the smallest value of N, where all edges of a complete graph of (N-1) vertices are colored either red or blue, and red-colored and blue-colored subgraphs should not form any complete subgraph of r vertices and b vertices, respectively. A superimposed red-colored and blue-colored subgraph is called a Ramsey graph. The parallel algorithm, which is based on the hysteresis McCulloch-Pitts neural network, found five Ramsey numbers successfully. The neural network requires n(n-1) neurons for a Ramsey number of an n-vertex complete graph. The simulation result for the algorithm is so promising that it may be possible to find unknown Ramsey numbers. >