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We show that the 2-edge-colored chromatic number of a class of simple graphs is bounded if and only if the acyclic chromatic number is bounded for this class. Recently, the CSP dichotomy conjecture has been reduced to the case of 2-edge-colored homomorphism and to the case of 2-vertex-colored homomorphism. Both reductions are rather long and follow the reduction to the case of oriented homomorphism in "Graphs and homomorphisms" by Hell and Nešetřil. We give an alternate proof of the case ofdoi:10.1016/j.ipl.2017.02.009 fatcat:q3qusqx7jjayfb4urvcjjy7c44