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We propose a class of new cutting planes to strengthen the Lov\'asz theta function, a well-known semidefinite programming (SDP) relaxation for the stable set and the graph coloring problems. For both problems, we introduce two new cutting planes that are derived from odd-cycle constraints and triangle inequalities and are valid for certain subgraphs induced by odd cycles. Computational experiments on graphs from the literature show that for each problem, one of the proposed cutting planes gives very good bounds.