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We prove Deligne's conjecture for central critical values of certainautomorphic $L$-functions for ${\rm GL}(3)\times {\rm GL}(2)$. The proof isbase on rationality results for central critical values of triple product$L$-functions, which follow from establishing explicit Ichino's formulae fortrilinear period integrals for Hilbert cusp forms on totally real etale cubicalgebras over $\mathbb{Q}$.

On Deligne's conjecture for certain automorphic $L$-functions for ${\rm GL}(3)\times {\rm GL}(2)$