Asynchronous P Systems for Graph Coloring Problems

In the present paper, we consider fully asynchronous parallelism in membrane computing, and propose two asynchronous P systems for two graph coloring problems. We first propose an asynchronous P system that solves the k-coloring for a graph with n nodes, and show that the proposed P system works in O(kⁿn²) sequential steps or O(n²) parallel steps using O(n²) kinds of objects. We next propose an asynchronous P system that solves the minimum graph coloring for a graph with n nodes, and show that the proposed P system works in O(nⁿ⁺²) sequential steps or O(n²) parallel steps using O(n²) kinds of objects.