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We relate coding for the two-user multiple-access binary adder channel to a problem in graph theory, known as the independent set problem. Graph-theoretic approaches to coding for both synchronized and nonsynchronized two-user adder channels are presented. Using the Tuŕan theorem on the independence number of a simple graph, we are able to improve the lower bounds on the achievable rates of uniquely and -decodable codes for the synchronized adder channel derived by Kasami and Lin. We are also able to derive lower bounds on the achievable rates of uniquely decodable codes for the nonsynchronized adder channel. We show that the rates of Deaett-Wolf codes for the nonsynchronized adder channel fall below the bounds. Synchronizing sequences for the nonsynchronized adder channel are constructed.