In the present paper, we consider fully asynchronous parallelism in membrane computing and propose asynchronous P systems for the following four graph problems: minimum coloring, maximum independent set, minimum vertex cover, and maximum clique. We first 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(n n+2 ) sequential steps or O(n 2 ) parallel steps by using O(n 2 ) kinds of objects. Second, we

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... ose an asynchronous P system that solves the maximum independent set for a graph with n nodes and show that the proposed P system works in O(n 2 · 2 n ) sequential steps or O(n 2 ) parallel steps by using O(n 2 ) kinds of objects. We next propose two asynchronous P systems that solve the minimum vertex cover and the maximum clique for the same input graph by reduction to the maximum independent set and show that the proposed P system works in O(n 2 · 2 n ) sequential steps or O(n 2 ) parallel steps by using O(n 2 ) kinds of objects.