This paper considers the problem of optimal subcarrier allocation in OFDMA systems to achieve the minimum outage probability while guaranteeing fairness. The optimal subcarrier allocation algorithm and the maximum frequency diversity gain are both analyzed through the maximum matching method based on the random bipartite graph theory. Accordingly, a surprising result is found, which shows that the maximum frequency diversity gain in subcarrier-sharing OFDMA systems is the same as that in point-to-point OFDM systems that serve only one user by using N subcarriers. It is then demonstrated that this maximum frequency diversity gain can be achieved by a proposed Random Vertex Rotation based Hopcroft-Karp (RVRHK) algorithm with the time complexity of O(N2.5), where N is the number of subcarriers. Because the theoretical analysis and the RVRHK algorithm are both based on the maximum matching method, the maximum frequency diversity in OFDMA systems is referred to as the max-matching diversity in this paper.